Question: Do yeast die after bottling? 

My second batch of beer have been stored in bottles over a month now without producing any co2.
I want to add some priming sugar to start the fermentation again, but I'm unsure if it will help.

Answer: 
Yeast will become dormant and eventually die after a few weeks to months, but only after any food sources, like priming sugar, have been consumed. 

The lack of carbonation after a month could be caused by a number of things

* insufficient yeast - normally there is yeast in suspension after primary, but a highly flocculative strain may settle out completely. For bottling, 10,000 cells per ml is considered a typical cell density.
* if the priming sugar was added to the entire batch, it may not have been mixed with the beer fully, leaving some bottles without sugar and others an excess.
* the bottle caps are not on airtight
* your temperature of 19-25c should be warm enough, but typically cold temperatures can hinder fermentation.

Adding more priming sugar can be done on a bottle or two as a test, which you can open after 3 days. You should hear a hiss, but the beer itself won't be carbonated unless it's left for another week. But if you get the hiss then the yeast are working it's most likely a problem with distributing the priming sugar evenly. No hiss, and you may need to pitch some more yeast. A tiny pinch of yeast is enough for the entire batch, but putting a tiny pinch of yeast in each bottle will work also.
Yeast are pretty hardy and it takes time before they completely die off. Even when the nutrients are nearly all consumed the yeast simply go dormant. I have cultured yeast out of beer bottles that were at least a year old. Granted the total viability isn't high but they aren't all dead. Alcohol levels, oxygen levels and storage temp plays a big role.

In your case, adding more priming sugar should work as I am sure there is plenty of yeast around. Be sure the beer is at 70F/21C before starting. You want the beer at room temp as the yeast see the sugar. For ease you may want to try something like Coopers carb tabs. Little pills of sugar for carbing one bottle at a time.