GPL is infectious, so my application would become GPL once a single GPL-licensed file is included.
Per the FSF FAQ,
you can combine code released under the other license with code released under the GNU GPL in one larger program...All GNU GPL versions permit such combinations privately; they also permit distribution of such combinations provided the combination is released under the same GNU GPL version. The other license is compatible with the GPL if it permits this too.
So to answer the question, yes.
However, MIT is compatible with GPL, and MIT isn't infectious, so my application doesn't have to be GPL.
That is incorrect. From the above quote,
...they also permit distribution of such combinations provided the combination is released under the same GNU GPL version.
However, in this case, the only GPL file I can see is
# Source: https://github.com/rstudio/shiny/blob/master/R/stack.R
# License: GPL-3
# Relicensed a MIT with permission.
I would contact the authors of rstudio, to validate this claim before assuming it to be true.