Since it's a hot topic to anyone following the us elections, do you ever apeace other civilizations? (accept "give me x or else" deals) I never, usually results in me vs everyone some time in the middle ages.
The only time I do it is when the demand is cheap (like a resource) and it'll buy me a 10 turn peace treaty to prepare my invasion fleet.
I do find it rather annoying that some Civs think they need to make demands on a regular basis, even when my power graph is above theirs. Catherine is certainly high maintenance at times, and cute when she pouts.
Weak civs, on the other hand, I like to strengthen, especially if they're in poor relations with one of the powerhouses. If a weak civ is at war with one of my rivals, I'll gift them military and other techs (this, of course, presumes no tech brokering option) so that they can provide some resistance, thus effectively enabling me to gain an advantage over the rival who is engaged in a war and not making as much progress as expected.
I wouldn't give a military tech to a powerful rival, but something like Liberalism or Economics (is that the one that grants Free Market? Whatever gets them out of Mercantilism), techs that I want them to have (unless I'm racing for communism, then I'll hold on to Liberlism) I'll let them have it and take the peace+diplo points, sure.
Everyone that has posted so far about never giving into ai demands should try playing Immortal or Diety. Lemme know how it works out for you.
I assumed from your first post that that you differentiate between value of various techs and base decision on that. I value ALL too much to give them to a powerhouse (more often than not I'm not friends with them) because even if not so valuable on their own, they need to be researched for something else that might be. "Build some universities and cottages, instead of countless troops and endless workshops if you want to research faster, no help from me", I think to myself. Probably the only exception is Mass Media, which I will gift to AP owner if it suits me to end the AP era (if not, I would avoid researching it anyway).