I often think that if I want to learn, I should think of draw offers as "offers to remain ignorant". Bobby Fischer was famous for turning down draw offers "out of hand" - without even thinking about them. Since experience is a major part of getting better, why would you want to deny yourself experience (unless you are not feeling well, or some other external consideration)?
The following is a true dialog, which occurred when I asked a friend why he offered a draw in a better position:
> "Hey, (ANONYMOUS), don't blame me, I don't know how to win this thing."
> "But if you are going to give people draws, how are you going to learn how to win it?"
There are three primary possibilities if you decline a draw (assuming no heart attack, etc.):
1. You play on and lose
2. You play on and draw
3. You play on and win
In all three cases you have probably learned something for your future games. Also, if you have a good position, then winning is more probable than losing, and your opponent might take a draw later if you throw away the win. If you lose, you are most likely to learn something, because then you made a mistake that should be identifiable. The goal should be
both to have fun and learn something.
Also, my teacher once gave me a draw when I was up the exchange in the endgame. I asked him "Why?" and he said "I didn't know how to win it" and I replied, "Except for the opening or endgame, do you really know how to do/win anything exactly? - you have to figure it out. And if you can't figure it out, try something and see if it works." In his position there was many ways to win, and the ones that didn't drew anyway, unless he put his Rook en prise, so he had nothing to lose and everything to gain (much more likely the latter).
In another case, a friend of mine said he always offered a draw to higher-rated players when he had a better endgame. I asked her "Why?" and she said, "Because I don't play the endgame very well" and I responded something like: "Hmmm. Let me get this straight. You don't play out good endgames because you don't play them very well. Could it be that you don't play them very well because you don't play them out?!"
On the other hand, there are some times you should accept the draw offer:
If external factors require you to leave soon, such as not feeling well, your ride is leaving, you are getting sleepy and need rest for tomorrow, etc.
If your game is clearly drawn in the endgame, then it is a little insulting to make a competent player keep playing until the 50-move rule, 3-fold repetition, etc.