Firstly, check that the connector is firmly seated into the socket and re-check the heater. You can also check to see if the LED - LED02 in the corner of the board, next to the extruder heater socket - is lighting when the heater is on.
If it’s still not working, we need to check to see if there is any output on the extruder heater. You need to set your multimeter to the DC voltage range turn on the heater and check the output. Please be very careful not to short out the probes when measuring the voltage. If all is OK, the reading should be 12 volts.
However, I have a nasty feeling that you will not get a reading there…
I have checked out the fuses on the RAMBo. There are 2 x 5 Amp fuses and 1 x 15 Amp fuse.
The 5 Amp fuse by itself is for the motors; I think it was the one next to the 15 Amp fuse which blew on your board?
The 15 Amp fuse is for the heated bed, whilst the 5 Amp fuse next to that is for the logic circuitry, fans and (most importantly) the extruder heater.3d printing service sla technology
As your logic and fans are working, then the fuse is also now OK. However assuming that the heater itself (as per my previous post) is OK, then this does point to a problem in the RAMBo output circuitry.
Unfortunately, these fuses are not particularly quick to blow; it seems that when you plugged the hot bed into the extruder heater output, both the fuse and the output circuitry were taken out at the same time.
The simple remedy is to replace the RAMBo board, but this in not cheap. A partial solution would be to re-program the firmware, setting the extruder heater on the hot bed output. You would of course, not be able to use the hot bed.