<p>During a print a lot of plastic ended up ripping the nozzle's yellow insulation strap. </p>

<p>Can printing without this insulation around the fusion chamber damage the printer?</p>

<p>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 </p>

<p><a href="https://i.stack.imgur.com/CYGI7.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CYGI7.jpg" alt="highlight of the insulation strap"></a></p>

No, without the insulation you can print without a problem. I've been printing for years with (cotton or silicone) and without insulation, works perfectly. Although heat radiates from the heater block, I've never experienced issues that it causes overheating of your printed parts.
Note that the insulation tries to contain the heat in the nozzle preventing heat to leak to the surroundings (less energy is used/wasted). As such you may need to do a new PID tuning (certainly when the print cooling fan is incorrectly positioned as such that it cools the hotend heater block), but printing should just work fine if the print cooling fan is in the correct position.