Well, that depends on the venue. In a small room, say one that seats around 50 - 100, it may be perfectly fine to screen using a DVD. I know Angry Filmmaker Kelley Baker has successfully screened (without technical problems) his movies quite possibly hundreds of times thus far using DVDs. I've also screened Date Number One about 20 times so far, to crowds that range from 25 - 100, using DVDs.
HOWEVER, some venues may have trouble using DVDs; not sure why this exactly is, 'cause DVD players work perfectly fine for millions of people - so, it is a good idea to have your movie on Beta or DigiBeta or some other pro tape format. Also, many film festivals may also request your movie on a pro tape format for their screenings.
Another option is to screen using MiniDV. I've tried this - using a MiniDV camera as the playback source, works fine. If you have a feature you'd have to get it on a MiniDV or DV tape that can hold it (i think some tapes can hold 90 - 120 mins.). Plus you'll have to make sure there is a MiniDV or DV deck, if you plan on doing several screenings; so that you do not wear out your camera by using that as a deck.
But, if you can't afford to create a pro tape version of the movie or you just don't want to do it, and specially if you are producing your own screening, DVD might work - based on size of the venue, distance from screen & power of the projector, desired image & audio quality, etc.
2. Filmmakers make movies, distributors distribute.
Distributors have advanced that idea; that if you have to do the "difficult or unsavory" work of distribution, you are not being a filmmaker. This is completely false, specially for DIY filmmakers; filmmakers should know everything there is to know about film distribution & marketing & have experience with it & be able to carry it out if need be, otherwise you maybe making movies just to watch at your house only.