Is my thermostat keeps losing power due to a furnace fault?
I will assume your thermostat is a digital one.  Digital thermostats can get power 3 ways:

From the C wire on the furnace transformer
From leakage current across the relay coil
From batteries in the thermostat

MOST of the digital ones on the market today have a C terminal.  However there are some that &quot;steal power&quot; these are intended for when you are using antique thermostat wiring that does not have enough wires to have a C wire.  Some of these had batteries some did not.
The batteries are for BACKUP in case of power failure.  NOT for a permanent source of power.
I suspect you do not have a properly hooked up C wire.  Your old stat had backup batteries in it and stole power off the relay coil.  It also lost power too but during those times it could not get power off the relay coil it used it's batteries.  Your new stat lacks backup batteries or it has dead batteries and your C wire isn't correct so the stat has to get power from the relay coil that is why it cuts out periodically.
Without more info we cannot possibly tell you how to troubleshoot your thermostat wiring.  I would normally suggest an HVAC pro but my experience reading the professional HVAC forums is a LOT of HVAC techs out there don't understand thermostat wiring anyway.  That is sort of understandable because furnace and thermostat designers seem hell bent on complexifying something that really should be very simple.  Take pics of the stat and supply it's model number, take a pic of the terminal block in the furnace that the stat wiring is connected to and make sure the pics show the wiring colors used on the furnace terminals.