I was thinking about powering my multimeter by Li-Ion battery using a boost module like this
Are there any problems in doing so?
Well, if you just plug it where the 9V battery was, it will run all the time, draining your battery quite quickly.
You would have to find a way to turn the converter on and off with the multimeter.
Besides that, I’m not sure how your meter will like switching noise where it expects a nice, clean battery.
That might cause some problems, depending on the actual meter.
In the end, it might not be a very practical approach.