Jinx was my most played last season and is number 2 right now in S5. I have tried a tear build and I see the synergy with her Q, it basically allows you to leave it on all game if you want, but I really don't recommend it.
Other items compete much better for that slot. A full 6 item build in order should be IE - Greaves - PD / Shiv - LW - BT / Bork - Defensive (QSS or Banshees). If you're feeling safe get a second PD as your 6th item. Every piece of this puzzle is pretty much required. Muramana just doesn't offer any meaningful advantage to be a better choice.
Snowballing. Jinx snowballs harder than almost any other ADC. This is true in specific teamfights, but also in a more broad perspective for the entire game. Putting that first bit of gold into a Pickaxe means you can play much more aggressively and secure kills to help you start snowballing. Tear really negates this possibility.
Rockets have a flat 20 mana cost. Early in the laning phase this can be rough, and having a tear definitely lets you spam rockets more right when you buy it - but as the game goes on that 20 mana cost becomes almost meaningless just via your base mana pool. At level 18 you have about about 1000 base mana. Take out 250 for a W E and R and you have 750 mana for Q spam. That's almost 40 auto attacks. 40 auto attacks is enough to kill everything and then some. Also consider that your mini gun form is better for killing tanks and cleaning up kills with its higher DPS. You only need rockets for softening people up and waiting for a safe moment to go ham.
Tear creates a cart before the horse situation. If you buy tear your autos don't do as much damage, therefore you need more autos, which costs more mana. The cycle continues. However, mana management is a skill that comes with champion mastery. The better you get at Jinx the less oom you'll find yourself. In the meantime grab some mana pots for your first few times back to base. Watch pro VODS playing Jinx and study when they use their mana.