KeratinPearls is pretty much right. I would say though that if you know the right people and express enough interest, you can land a competitive (GI, derm, GU) fellowship without a publication. You would be strongly advised to have at least abstracts though. You would probably be unlikely to get a competitive fellowship at an outside program (i.e., one in which you did not do your residency or do a significant visiting rotation) without some abstract/presentation/publication on your CV.
For some of the less competitive fellowships like cyto and general surg path, if you are a solid resident with good references from a solid program, you don't really need any of this. Having it is always better though.
I would caution you against outright "avoiding" research. Obviously not everyone does or should do bench research. But participating in a couple of projects during your residency, perhaps presenting them at meetings, does a lot of good towards your future career and your understanding of the field in general. Personally if I was hiring someone for a job, which I might do someday, I would be hesitant about any candidate who did absolutely nothing academic during their training. It shows your curiosity, it exposes you to different types of thinking, things like that. Even if you only do it for selfish reasons you are likely to learn something. It's kind of like volunteering when applying for med school.
Ultimately though, the answer to your question comes down to three important questions:
1) Does your program have fellowships available and do they consistently take internal candidates?
2) What fellowship are you talking about?
3) What kind of resident are you? (i.e., responsible, innovative, intelligent, vs lazy, doing the minimum, needs to be told things 3 times, etc)
