Red seems to have some effect on performance in well constructed experiments and in sport, though we could do with further studies
The trouble with many studies on the relationship between colour and performance is that they are based on speculative theories and rely on poorly controlled experiments. This is summed up by the authors of this paper in the Journal of Experimental Psychology thus:
...in the literature at present, there is no clear evidence for a color effect on performance attainment. It is possible that this summary statement reflects the fact that color effects on perfor- mance attainment do not actually exist. However, it is also possible that color–performance relationships do exist but that weaknesses in the existing research have made them difficult to detect. Several important weaknesses may be noted...
This papers report the results of carefully controlled experiments in a variety of environments on whether the perception of red alters performance in intellectual tests such as IQ tests or solving anagrams. And they find a significant effect. As the summary says:
Four experiments demonstrate that the brief perception of red prior to an important test (e.g., an IQ test) impairs performance, and this effect appears to take place outside of participants’ conscious awareness. Two further experiments establish the link between red and avoidance motivation...
So, in simple terms, red reduces performance but not in a way people are conscious of. And one of the mechanisms is an unconscious avoidance of harder tasks when given a choice.
This clarifies part of the possible mechanisms behind the observations in sporting contests: it isn't that wearing red makes you perform better, it makes your opponent perform worse. The classic paper on this is a paper in Nature (abstract only online, but, for readers in the UK the British Library is your friend) from 2005 (though it assumes that red makes the wearer perform better). The paper uses the fact that some sports randomise colours so removing most of the possible confounding factors from the statistics. Speculating that red may have some residual effect as a primate dominance signal they report significant differences between player performance when players wear red in 4 olympic sports from 2004 (boxing, tae kwon do, greco-roman wrestling, freestyle wrestling). As they report:
we found that for all four competitions, there is a consistent and statistically significant pattern in which contestants wearing red win more fights
They find an excess percentage of wins of between 2% and 5% over the 50% expected if colour had no effect. This effect was >10% with evenly matched opponents but not significant at all for opponents of significantly different abilities (which is what you would expect for a small effect).
But does this apply to sports like football (that's soccer for americans)? In 2008 a paper appeared in the Journal of Sports Sciences (again abstract only online) that used a large dataset of football results from football in England from 1947 to 2003. They used several approaches to attempt to disentangle confounding factors. For example, one proxy for teams support base (which might independently relate to capability) is city size. They report that in most cities with more than one team consistently in the top leagues it is the one in red that does better. Another factor that matters is the observations that teams often don't wear their home colours in away games. They report:
Across all league divisions, red teams had the best home record, with significant differences in both percentages of maximum points achieved and mean position in the home league table...No significant differences were found for performance in matches away from home.
The obvious weakness in their analysis is that they don't analyse away matches by what colours were actually worn (teams will only swap from their home colours when there is a clash). This might significantly improve the statistical quality of the analysis as would updating it to cover the last 8 years. Also, there are now large statistical databases of team performance which have some value in predicting results (see here). Combining these with accurate colour information in both home and away games might improve the statistics for recent games and highlight the size of the effect. I'd love to see other international leagues analysed.
So, in summary, wearing red seems to reduce, by a small amount, the performance of your opponents in sport and the effect of seeing red also seems to operate to lower performance in non-sporting activities.