To answer this question, I gathered data from 2010-2015 and looked at how teams performed in the game following a non-standard week meaning they played on Thursday Night Football or Monday Night Football, or had a bye week.
Starting with Monday Night Football, if the short rest had an effect, we would expect to see a winning percentage significantly less than 50% for the game the following week. Over the 193 game results I gathered, the team playing on short rest went 91-102, a win percentage of 47.2%. A chi-squared analysis gives us a p-value of 0.6048. In other words, if the games results were determined by the flipping of a coin, we would see this much deviation from the expected win percentage about 60% of the time, meaning we cannot come to the conclusion that teams perform significantly worse on short rest. For the results to be significant, at the 5% level, we would have had to conduct the chi-square test using an expected win percentage of greater than 58% or less than 38%. Given the large sample size, it is reasonable to assume that teams coming off of Monday Night Football would’ve averaged out to about a 50% win percentage, but even if it was slightly different than 50%, the chi-square boundaries are far enough apart from 50% that the conclusions should still hold.