I kind of feel for Tina here. Yes, she's being silly and irrational, but all of us have times when we're silly and irrational about things or people we care about, and she has the added hell of having this play out on vacation in front of a bunch of other people.

Bottom line: Tina is the Zeppo.

Links aren't allowed on /r/relationships, so I'll condense the explanation as much as possible. There's a really famous episode of Buffy the Vampire Slayer titled "The Zeppo," and it's a reference to the fifth Marx brother that most people barely know existed. The Zeppo is the person who does not have an obvious role or place in a group. Scratch that; they do have a role, but the worst possible one; he/she creates or cements the "in-group" by becoming or being the "out-group." In essence, you get to belong to something because they so clearly don't.

From one perspective, this is crazy because none of us have "jobs" in a social group when our only purpose is to be friendly and enjoy each others' company, and she's ruining your vacation for nothing. This is your point of view on the proceedings and you're right; Tina's being a pain in the ass and it all seems to be over something that, while regrettable (Rob doesn't love her back), she needs to deal with.

But from another perspective, Tina's kinda right too. Rob and Janet have now paired off and become the second couple on a trip with an existing couple (you/your wife). She has been converted from person #4 to person #5 on this trip and no one seems to have objected to this; if anything, from her perspective they've been actively helping it along. Nobody enjoys being the Zeppo.

There are some things in your post that jumped out at me. They are good examples of how a desperately insecure and unhappy person is unintentionally made to feel even more insecure and unhappy by the people around her:

• Tina is really nice but a little overweight. She's insecure about her looks (which I notice you don't comment on) and feels wretched when she can't keep up with the more sporty activities that Rob enjoys. Bonus: With the good-looking, super-fit Janet "unseating" her, she gets to wonder if Rob would have gone for her if she'd gotten it together and lost the weight.
• Then my wife and Tina joined in and they started talking about politics and Rob and Janet again, found a lot of similarities as did my wife and they really seem to be enjoying themselves. This entire time, Tina was sitting there glaring in their direction and sulking. Translation: Tina doesn't have an obvious ally in this group. Everyone hits it off with Janet. You talk to her on Facebook and like her. Your wife likes her. Rob obviously likes her. Tina's role in this previously four-person group is being threatened on multiple levels. Is she person #4 here, or person #5?
• Rob was talking about how Janet was going to take him to a gaming bar on Friday. Janet did not invite everyone: She invited Rob. Tina's sense of being excluded begins, even though -- yes -- it's kinda crazy because she isn't into gaming and by all rights shouldn't care.
• So she turned up in exercise gear and then Tina decided she wanted to work out too despite her never having had a gym membership. Tina doesn't really want to be excluded even though she isn't into working out either. But what she wants and who she is are fundamentally incompatible. Unrequited love is a real bitch.
• So Rob said that she was going to make Janet late for her class but Janet said that she could take the class another time and just workout normally, she was happy to wait for Tina to get changed. Janet is actually doing the right thing here by disguising what is actually a romantic advance as a social invitation. I mean, it's a vacation, isn't it? Vacations aren't about trying to split people up or exclude them; everyone who wants to come should be able to come to the gym! But Rob hasn't gotten the message, and is kind of being a shit by not attempting to disguise his own motivation for going. He doesn't want to go to the gym with Janet and Tina: He wants to go to the gym with Janet alone. He's interested in Janet, but Tina's an unwanted and unnecessary addition. Tina would have to be extraordinarily dense not to pick up on this.
• After the gym, Rob and Janet got lunch together Of course they did.
• To be honest, the dinner after that felt more like a double date. It was a double date. Tina knew it before Rob/Janet were publicly admitting their interest or you/your wife realized this was serious. Does anyone out there like being the fifth wheel on a double date? Because if so, I haven't met them.
• My wife is running out of patience, especially since Janet at my wife's request has taken time of work and she asked us, and now a friend of hers has also joined us from an other state. This trip is rapidly becoming Tina's nightmare. Your wife has invited the person who is making Tina feel excluded and insecure to spend more time with the group, and even asked her to take time off work in order to do so. From Tina's perspective, it's not just Rob anymore -- the whole damn group is against her. I don't think your wife did this intentionally, but the effect was the same nonetheless. But after all, you all really like Janet, right?
• If she had said she didn't want to go from the get go, we could have just taken a regular car. Zeppo-status accelerates! Now the whole group's pissed at her because: a). Text: She's caused you to spend more money on a larger car even though now she doesn't want to go, but: b). Subtext: She doesn't want to go because she doesn't want to hang around the person who makes her feel like shit.
• All this is really turning Rob off Tina. Rob would have found a reason to get pissed at Tina anyway. He doesn't want her hanging around making a pest of herself while he's trying to move things forward with Janet, even though a vacation's really not the nicest time to do that. So, adding this up: You're annoyed with Tina because she's mopey and sulky and stays home. Your wife is annoyed with Tina for the same reason. Rob is annoyed with Tina because of the aforementioned reason. Everyone seems to prefer Janet and even her friend. Tina is probably aware of all of this but is still in love with Rob, knows he doesn't owe her a relationship, knows he hasn't really done anything wrong here, feels like shit around Janet even though Janet doesn't do that purposely, feels like shit for annoying everyone else but can't help it. There's a lot of shame and self-loathing that comes with compulsive behavior.
• He's asked her several times what the issue is Rob, buy a clue.
• Rob spent an entire day taking Tina shopping was bored out of his mind whereas with Janet and her friend, they took him to a local "footy" match and recommend or takes us out to some pretty nice restaurants. I bet Tina totally never picked up on Rob's boredom on that shopping trip. I bet he disguised it really well. I bet he didn't evince glee or relief at the thought of being able to do something else that didn't involve Tina.
• They also went on a hike together with Janet's friend because they are fit and then Tina went with them and couldn't keep up, so they slowed down for her but Tina got sick of it and turned back so they offered to turn back too but Tina just decided to pay for an Uber home while they finished the hike. This is perhaps the biggest point of humiliation for Tina. When you're out of shape, it's legitimately embarrassing not to be able to keep up with people who are in good shape. You feel sweaty and disgusting and you're panting like crazy in front of people who you know are probably irritated at being slowed down. You know they could be covering a lot more ground if you weren't there, and they know it too. Emotionally, Tina feels like the Zeppo and now she's even physically the Zeppo. She literally cannot keep up with them. Getting in shape will take time and will not solve any of her immediate problems.
• To make things worse, she's unfailingly polite and nice to Tina, even when Tina is being snarky, then Janet and Rob exchange "a glance" and Tina just feels even worse. Well, yeah. This stuff tends to serve as a vicious reminder to the out-group that it is the out-group. The in-group is united in its exasperation over Tina's behavior, to the point of being able to communicate non-verbally on it. It's kinda cruel for Rob and Janet to do this, especially as they've known each other for, what, a few weeks? Tina's conversion from in-group to out-group was remarkably swift, wasn't it?
• Tina is even refusing to go to things with Rob or my wife and I even if Janet is not there which is stupid Because it isn't just about Rob -- not anymore. You've admitted the whole group is annoyed with her, and there's no way she isn't aware of this. From her perspective, why inflict her unwanted presence on anyone? She's gotta be wondering: Did anyone actually want me along on this trip, or did they just bring me to be nice? What are they saying when I'm not with them? I've known all of these people for years and I thought they liked me, but they dumped me the moment they had someone they liked better around! This stuff is keeping her up at night and she's counting the days until she goes home (although now Janet's thinking of moving to Canada, sooooooo ...).