I've noticed that if I don't reset my watt meter once per sometime, which I use to measure mains voltage, it completely stop responding. The timer stop working and the buttons not responding.
If I then reset it, it returns to work.
Why this is happening?

Likely a bug in handling timer counter overflow (or better to say lack of proper handling). Most likely just never tested for operation longer than a day or so...

Usually it takes several weeks or a month to stop responding.

Could be. E.g. WinXP crash after 49 days, when uint32 ms counter overflows. Newer wins patched that by moving into a 64bit variable.
And I intentionally write "patched" and not "fixed", because although a 64bit variable counting ms seems to expire only after half a billion years, at some point when computing power would become way bigger someone may redefine it to e.g. counting picoseconds and the overflow problem would be back, that time in a less than a month.
The only correct way according to me is to define it as a revolving counter, so all SW has to count on its cyclic nature. Because it is not needed for anything longer than a few seconds, so overflow in hours or so should be rather easy to manage.
In the wattmeter it may mean transition from an "average since power on" into "low pass filter with few month time constant", all that without any need for any extra data memory space.