Should I keep iPhone plugged in all day or is that bad for it?

No, you can keep it charged all day if you want. It uses a trickle charge with a sensor that monitors the actual charging. Lots of people keep it in the charger whenever not out and about with no detriment.

You can keep it plugged in as long as you want without damaging it. it knows when to go into trickle charge to keep the battery topped off.

Glor

Just to expand on Ron's and Glor's answers, not only is leaving it plugged in frequently not bad, it's actually good, in that you'll put fewer cycles on the battery over time, which will prolong the battery lifespan. Lithium batteries do need to be used to maintain health, but the best usage is frequent and light (i.e. few deep discharges). Leaving it plugged in most (but not all) of the time is beneficial (just not necessarily practical for most mobile phone users).

Any lithium battery is good for 400-500 charge cycles, or ~3 years, whichever comes first.

You can learn more from the following links:

http://www.apple.com/batteries/
http://www.apple.com/batteries/iphone.html