'I have a normal programmable thermostat. On the furnace they installed
a "timer".

I think by 'timer' , you mean an automatic Time Delay Relay (perhaps ?)
whereby the signal to the 2nd stage is delayed for 10 minutes.

'After the thermostate turns the furnace on, it will run at low heat (3
burners) for 1 to 10 minutes (depending on the manual setting of the
timer). After that time it will kick into high heat (2 more burners). '

From this description, it would only be possible if you had a split
manifold in your furnace with one gas valve controlling 3 burners, and
another gas valve controlling the remaining 2 burners. I doubt this is
what you have very seriously. Typically on a 2 stage gas furnace, you
would have ONE gas valve that works in 2 incremental stages : the first
stage produces a smaller flame thru ALL burners...then, when second
stage is demanded, the same gas valve would let more gas into the ALL
the burners thus increasing the flame (and hence , heat output) until
the thermostat is satisfied then all gas is shut off to the burners. A
2 stage setup is advantageous , especially in areas where there is a
significant milder winter period . Personally, a 10 minute time delay
before 2nd stage is engergized..is too short of a duration ; 15-20
minutes would be better to give low stage a chance to satisfy the
thermostat.