How *safe* are cells in series?
With good batteries, it will work fine and it used to be a common practice. Many tube schematics label the high, positive power supply for the plates as "B+". This stands for "Battery positive".

Batteries can cause death by electrocution or by starting a fire. I recommend using a fuse and taking high-voltage safety precautions. See Tubelab Electrical Safety.

The idea has challenges, but no worries, I have a suggestion below that you will like better: 


It is not great to create a waste stream of dead batteries
Expensive
Mechanically challenging to make reliable
Any one failed battery ruins the whole pile


Many cheap CR2032 batteries are actually 'recycled' (dead), even though they have authentic-looking packaging. I have had this problem when buying from my favorite normally-reliable online store.

An easier way to build a high-voltage battery is to chain a bunch of 9V batteries together. 9V batteries snap right together in series, and are good for a quick check of the power supply requirements for your circuit. When it is time to build something better, you can measure exactly what you will need.

There should be no problem driving the IN-14.

Take care, they pack a punch!