Strange circuit.
R2 and R3 having different values means current sharing for power dissipation can't be purpose, and whatever that purpose is isn't obvious to me.
Badly depleted batteries would cause max current from LM317 to flow, limited by 1.6 A fuse.
Current in R1 will roughly be total current flowing in R2 R3 pair divided by hfe of Q1. The hfe of the TIP-122 is stated at 1000 so current in R1 will be so much less than the current in either R2 or R3 so it shouldn't burn at all, let alone burn badly.
Vce sat of the TIP122 is stated at 2 volts given 3 A, so won't be much less than a volt at the max operating current as set by the fuse.
Since this charger is operating as a float charger the current supplied should be relatively small. It is a compromise between battery life and recovery time when batteries are discharged. A 0.1 C charging rate is often used. For your 6 AH batteries this would be 0.6 Amps.
At 0.6 A the drop across the R2 R3 pair is about a quarter volt. Add to that four diode drops (two diodes and the two BE drops in the Darlington transistor. ) and you have about 3.2 volts subtracted from the difference between the LM317 and the battery nominal voltage. About 0.3 V across R1. This voltage needs to drive 0.6A/hfe or 0.6 mA through R1. So R1 would nominally be 500 ohms.
Higher current could flow in R1 if Q1 has failed open. Have you checked this?
But with a 500 ohm R1, even if Q1 failed open R1 would not burn without burning R2 and R3.
So something doesn't make sense. Changing the recharge rate to 1 C only drops the value of R1 to 50 ohms, still making it burning without burning R2 and R3 unlikely.
In spite of this confusion on explaining the fault, I suspect the circuit will work well enough with a resistor in the 50-500 ohm range. If you use 50 battery life will be compromised a bit, but it may be worth it for the faster recovery.