Some will, yes. Just do your research if you're not sure about who is running it. Ask around. Google.
Yes, an entry fee is no indicator of a scam unless it's out of tune with the prize offered. Check Poets&Writers website - very good source
also check the contest listings/warnings on preditors & editors... http://www.invirtuo.cc/prededitors/pubctst.htm
I've never found a writing or drawing based contest that required an entry fee. It's likely a scam, unless the prize is a very hefty sum of money.
^^ No, there is frequently a small fee to be paid, it's not necessarily a scam. Just check out the magazine/organisation to make sure it's legit. Random websites sometimes hold comps but can fold taking your competition money, though, so be careful if there is no registered address for an organisation.
Thanks. I came across a contest a few weeks ago with an entry fee, but I didn't get a good feeling about it.
even the most prestigous writing competitions have an entry fee... as advised above, just be sure you check out any you consider entering carefully, before entering, to make sure it's legit... first check them on p&e, then google for feedback and check out their website... if they don't have one, or you can't find an actual land address for them [not a mail-drop], don't enter...
I recently entered a writing contest and the entry fee was $8, all completely legitimate. Just depends on the market, as others have already noted.
Some off-topic advice. If you want answers, it's a good idea to make your thread title a bit more specific and meaningful.