Statisticles! as Bold Tone would say;
Researchers have shown that sacking the manager has no discernible influence on performance.
Here, here, and here. Got the links from a good sports economist here.
Interesting/funny quote from one of the abstracts that studied Dutch football:We find by analysing 12 years of football in the highest Dutch league...the improvement in performance after appointing a new manager does not exceed the seasonal average of both the old and new manager. More importantly, using a control group, it turns out that when the manager would not have been forced to resign, performance would have improved more rapidly.
People would retort: If it is not the manager, it's the players. But Wenger buys the players, and his players aren't performing, ergo, he should be sacked.
This would dovetail well with another statisticle laced article (though focused on American sports, of which the MLB is most like the PL in terms of spending restrictions, i.e. there aren't any) which looks at winning strategies depending on spending restrictions:

The article is good in that it explains that while the highest spenders don't win every year, on average, they win way more. People would retort: Well we do have a self-imposed salary/transfer cap because of the stadium/self-sufficiency principles.
TLDR; Sacking Wenger probably won't do anything. Spend the money Wenger!