as someone very similar to your boyfriend, i am absolutely mindboggled by all the comments here.
i'll digress and tell you my personal reasoning for it, which may be completely different from your boyfriend but it's just some food for thought:
i have had chronic moderate stomach pain for over 10 years now - since i was 13, i am now 24. i have been to over 20 doctors - regular physicians and specialized endocrinologists, i have had surgery when i was 18, i have tried every 'diet' there is (in the case it was some nutrient or whatever that reacted poorly within my system, e.g. gluten-free), it has never been properly diagnosed. it has definitely ebbed and flowed in severity over the years, but for the most part i live my days with a dull ache in my stomach which i've mostly grown used to but sometimes it's more of a severe, painful ache and i just lay in bed with painkillers until it subsides.
because this pain usually occurs after i eat, and since having a proper balanced diet did not remedy this issue, i usually eat twice a day - scrambled eggs in my morning, and a vegetable soup in my evening. whenever i go to a restaurant or wedding or whatever, it's always chicken fingers and fries. people act incredulously when they learn that my diet is 90% those 3 things. it's partly because i feel like those 3 things have the least chance to cause more stomach pain, and partly just because that's what i enjoy eating and i have a utilitarian "why fix what's not really that broken? (probably not the best adage since i am kind of broken, but whatever)" outlook.
some of these comments are really ridiculous to me. you read the typical "my gf just gained 30lbs and i'm not attracted to her anymore" posts and half the time you get the typical "lol get over it people gain weight when they grow up." no one cares if someone eats unhealthily to gain weight, but if someone eats unhealthily AND strangely - it's a humongous deal and they're going to get scurvy and die? i'd be really surprised if most of the people commenting how unhealthy this diet is, actually has a proper, well-balanced diet. is this type of diet healthy? no, he should definitely be taking vitamins and other stuff, i'm not anything near a nutritionist so i would be ignorant to comment further. however i've seen a lot of posts where people discuss things that are unhealthy (drugs, a sedentary lifestyle, never exercising, etc) but don't have comments that look as adverse as these.
"i've eaten more than that in the last 24 hours!" - ..so? i'm not saying what he's doing is healthy, but it's completely irrelevant that you eat a wide variety of foods when you can have a similarly unhealthy diet that just doesn't look as odd when put into text.
"i ate the same things as him and got sick" - if that's all he's been eating for years and it DOESN'T get him sick, it seems safe to assume that your bodies are working differently and what gets you sick may not necessarily get him sick. i'm sure he wouldn't eat those things if they had the same adverse affects on him than they do you.
"how are you going to RESPECT someone who refuses to eat a balanced range of foods" - what the fuck? me being a 'picky eater' doesn't mean i'm worthy of respect? would you say the same thing to someone who is in shape but doesn't exercise the recommended number of hours per day or drink the recommended amount of water?
"you can't go try new restaurants" - you would really break up with someone because they don't wanna go to a sushi place with you? i personally don't really get foodies - i eat because i'm hungry, i don't have a lifestyle based around it - but it's not like grilled cheese is a rare food.. you can get it at 90% of restaurants unless they sell very specific cuisine, and even so it may be available on a kid's menu.
if eating different foods and stuff is super important to you, or you base your identity around the things you eat like being a super strict vegan who mentions it in every conversation, then you guys are incompatible, but to me it just seems like you find it weird that he has a small palate and you're asking if it's cause for concern. yeah it's unhealthy, and i definitely recommend he start taking vitamins or add in some apples or bananas or whatever. he could also just do it out of convenience and it could be a phase - my brother used to eat nearly exclusively fast food, even multiple times a day, in middle and high school (he was very athletic and had a good metabolism, was never even remotely close to being overweight), and now he'll eat virtually anything. but really, the comments are making this out to be super shocking. the sky is not falling, your boyfriend is just a picky eater of unhealthy foods. he may lack energy, or gain weight as he grows older, or whatever. but many people don't have good diets. this is not in the realm of being a heroin addict or him being an alcoholic or him eating enough to gain 30lbs in a few months or whatever, and i've seen threads on those issue get less harsh responses than this one.
really just ask him if it's out of convenience or he's a truly picky eater. suggest he takes a multivitamin or add some fruits every once in a while. if his habits don't change, it'll likely cause problems when he's older. but i think many people here are overreacting, even moreso than r/relationships normally does :p