1: Publication pressure. There is too much emphasis on creating publications. The end result is that people rush out unfinished work, or end up diluting their findings by unnecessarily spreading them across several papers. People are also discouraged from undertaking projects that may take a long time to "pay off" with a publication.
2: Scientific networking. In scientific fields, people get to know each other, and those relationships still heavily influence the process. Fame and reputation matter when getting published, not to mention friendships with publishers or reviewers.
3: Bias towards excitement/positive results. This, I believe, is not entirely the fault of the scientific community. If left to their own devices, I think there would be plenty of scientists willing to double check old findings, or publish negative results. However, it's hard to convince people to pay you or employ you if your work is perceived as "boring," "negative," or as some sort of failure.
There are a handful of smaller issues (e.g. excessive emphasis on "impact factor") but I suspect that they are mostly symptoms of the above issues.
I've always believed that the whole "publication model of science" is due for review. It seems to be a product of the 18th century rather than something we'd decide was a good idea today. If we do it right, I feel like the positive results bias would immediately go away.
I don't think we can "fix" issue #2 in any really meaningful way. Even if you were to take people's names off of publications, they would still chat with each other at meetings or through collaborations. I think the best thing to do is not try to prevent people from influencing each other, but simply to make the whole process more transparent.