> told me that if I left the transformer plugged in it would burn up.
There's a big transformer on the pole in the street, powers my house and the guy across the street. Powered-up 24/6.99 (two outages this year). Been there 20+ years. Others are pushing 40 years. Mine hasn't burned up yet. Very old ones typically (and rarely) burn on a very hot day when the power company has not allowed for increased use of A/C, not from idle or light-load.
There's a 24V transformer in my furnace, keeps the thermostat "alive" so the heat will come on when needed. That one is only a year old, but the one before was 17 years and working fine when removed (firebox rot). One in my last house was over 25 years.
My Palm Pilot charger is a transformer and I just realized it is going on 15 years non-stop. Until recently all answering machines used wall transformers powered-up all the time.
With very few exceptions, 120V transformers SHOULD be fine powered-up all the time, for the "life of the device".
They won't live forever. All insulation eventually breaks down. They typically out-live the device they power (my Palm is way past retirement age; answering machines die of lightning or get replaced by a telco answering service). But I have had PTs fail, and worst-case is they fail low-R and burn up, or zero-R and burn the line cord. A flame-resistant box and a primary fuse are wise precautions (unless you know there is a Thermal Cutout inside).