Nozzle insulation gone, can the printer be damaged? During a print a lot of plastic ended up ripping the nozzle's yellow insulation strap. 

Can printing without this insulation around the fusion chamber damage the printer?

If there is no chance of damaging the printer, how likely is it that the prints will be affected by the absence of this insulation
Removing the insulation will not damage the printer, although it may affect the quality of the prints. The insulation is, after all, there for a purpose.

Allowing too much heat to escape radiatively from the heater block will reduce the maximum flow rate, since less energy will be available to heat the filament, and you may need to reduce your printing speeds or increase your nozzle temperature.

A bigger problem is that heat radiating from the heater block can cause already extruded filament to sag, especially when printing details and intricate infill. If the nozzle remains too long in any area, already extruded filament is likely to soften and deform. Without adequate insulation, the only way to counteract this is by increasing part cooling, and this will remove even more energy from the heater block.

Either fit new insulating tape or buy some silicone socks, since they are readily available for MK7/8/9 heater blocks. Then print a couple of Benchies, both with and without a sock, and compare the results.