I prefer to play Teemo as an AP with Attack Speed items such as Nashor's Tooth and I was wondering what his role is in a teamfight. He doesn't do a lot of damage with anything but his mushrooms. So what should I do during a teamfight? Play like an AD carry, an AP carry, or make a escape route if we start to lose? Also, should I concentrate on one person or spread my poison around? And should I use mushrooms to hit lots of people or save them for running? I have to be careful as I'm very fragile.
There are several ways to play Teemo (I've seen pure AP, AP/AS, AD carry, support, etc.), although the different styles often have several points in common.
Things you want to do regardless of build
AP Teemo in particular should drop shrooms as often as possible outside of combat, they have great scaling with AP and more vision never hurts. However, the small cast range on shrooms means that shrooming during teamfights will generally be a defensive maneuver. Be sure to drop one or two under your AD carry if you suspect that you will be initiated on.
The blind is one thing that I cannot stress enough. I have seen countless Teemos blind the enemy AP carry in an effort to score a kill while the enemy Jax or Fiora rips holes through the team. Do not waste your blind.
You won't be using your shrooms all too often during teamfights unless they engage on you. However, if you notice an enemy standing open in one spot (generally the AD carry), feel free to drop a shroom if you think you can do without too much risk. AP Teemo's damage during fights will come mostly from sustained damage on his Toxic Shot and burst from his Blinding Shot.
I disagree on this point. Teemo's damage, especially in an AS/AP build, is very significant. Play him like an AD carry, attacking whomever you safely can. Remember to use blind as well as poison; focus on blinding the ADC when possible, but a bruiser assaulting your carry is also a good choice.
Focus on whoever you can safely hit.
Since you're building AP, you want to hit as many people as possible. If you find your mushrooms getting picked off from oracles/pink wards, it may be possible to sneak in and drop them right at your opponents feet.
Consider building a Frozen Mallet on Teemo. The perma-slow and extra health compliment his existing kit very well.