Should I participate in non-funded bioinformatics research?
The fact that work was funded says almost nothing to me as a reviewer of a paper about that work nor about the odds that such a paper is good. I think funded work is more likely to lead to papers, because the researchers have been paid directly to work on it and have better odds of making progress. That being said, funding is not a guarantee of goodness.

Your assertion that unfunded work is normally weaker and unreviewed is not well founded. Unfunded work is often preliminary work designed to prove the feasibility of an idea and lead towards some sort of publication (poster, presentation, or article) to bolster the case for later funding. This is very important work and is often done by researchers (establish and new) who are moving into a new area or working on a new direction or idea. To attribute lack of funding to weakness is fallacious.

Also, many, many good ideas go unfunded. The rejection of a grant application is not necessarily a reflection that the idea is bad, wrong, unsound or otherwise unworthy of funding. Sometimes, many times, it represents the fact that grant programs are highly selective and very competitive. Some have a less than ten percent funding rate. This means that many good applications will be turned away. 

I would suggest that you soften your judgement about projects based on their funding or lack thereof and focus on which you think is a better idea for purely scientific reasons. If you truly get to pick, then I would suggest that you work on the one that's more interesting to you, will better promote your career, or may lead to the biggest breakthrough.