I've heard that Google won't index a site or a page that has a pop-up.
 Is this true?

Hi jxlaurie,

Good day and thanks for your question. 

I don't know where you heard this information, but I can assure you it
is 100% incorrect. 3 of my websites have pop-up ads (one has 4
pop-ups) and all 3 are listed on google with no problem.

://www.google.com/webmasters/ contains information on how pages
are listed and also dropped.

If you have any clarifications, pleae don't hesitate to ask.

Thank you for using this service and have a nice day

Warm regards,
aditya2k

Maybe the misunderstanding is, google does not like pop-ups ( along
with the vast majority) I maybe wrong but google does not allow
pop-ups if using adwords.

But it is quite true there will not be an indexing prob for sites that
use pop-ups