Although I'm not a professor, my position has a lot of similarity and I have often been asked to write recommendation letters. My reaction has been a mixture, depending on who is asking for me to write the letter:
- Students who I like and respect: "Sure, I'll be happy to. What are the aspects of your work that you'd most like to have emphasized?"
- Students who I haven't got a strong opinion on: "I'm not sure that I know your work well enough to write a strong letter. If you really need me to, I can, but I think you'd be better served by getting somebody who knows you better."
- Students who I have been disappointed in: "I'm sorry, but I don't think that I would be a good person to write the type of letter that you need."
You only want letters from the first type, who will happily sing your praises. And for a student like that, it's no burden at all: I'm going to want to push your career forward, because I'm hoping I'll get to see somebody who I like prosper, and maybe even return to work with me as a colleague in the future.
As for the multiple letters question: don't worry about it unless you're applying to like 20 schools. If the programs you're applying to are similar, they're probably just going to be writing just one template letter about their experiences and filling in a couple of blanks to customize for the institution. Now certain institutions have terrible forms that will make it a pain to send that letter, but that's hardly your fault, and your recommender will be used to it...