When using a flask application does using print negatively affect performance?
Both outputs (print and logger) end up in a file, the only difference is that a logger typically allows for some filtering of the output and tries to add as little overhead as possible when messages are filtered out. This means its not possible to say without testing, IE comparing the two. You should be able to use the timeit module to time how long it takes to execute a for loop that prints a thousand or million messages.
However, the other issue is concurrency: if your flask app is run by separate clients, the performance impact of log vs print may be different depending on how the log is designed vs the capture of print statements. This is harder to test but my guess is you could create a script that uses the multiprocessing module to run a wad of requests in parallel to a flask URL that just has one print/log message, and meadure how many requests your test client is able to make in a given amount of time.
The final consideration is that the amount of processing in your flask view/render may be significantly larger than time it takes to either log or print.  Eg if your render without any prints takes 100ms, and with print it takes 110ms, then what does it matter that log is twice as fast as print (IE 105ms instead)? As long as you remain frugal in your use of print/log, it won't matter much. This always applies, in any app, not just web.
I've deployed an flask app on heroku and since it's relatively small I never bothered to setup the logger. I have found by using a print statement I can see everything I need on one of herokus add-on loggers. What I'm wondering is does that negatively affect performance vs had I just used the built in logger in flask? Does the print statement add unnecessary overhead to a python program?