What is this, and can my dryer work without it?
The item referred to as a "rope" is likely to be either the drive belt or the felt strip that the drum rides on. 

If the dryer drum still turns when the unit is turned on then what you pulled out is not the drive belt! On the other hand the belt could have torn in which case the motor will no longer be able to turn the drum. Clearly you would not be able to ignore and run the dryer without the belt. 

The felt strip is typically fitted around the front periphery of the drum where it fits into a round track on the front housing of the dryer. The felt acts as a low friction bearing surface for the front support of the drum. If this is what you pulled out of your dryer then you do not want to attempt to operate the dryer without replacing the felt strip. To do so would lead to more serious damage as the drum became a metal to metal or metal to plastic bearing surface in the drum track.

For most dryers the replacement of the drive belt or the felt strip is relatively easy. You typically have to remove the front loading door assembly and the front housing of the dryer. (There may be other pre-steps to get to the point of being able to remove the front housing). Once you have access to the front of the drum it is easy to slide a new drive belt around the drum and onto the motor shaft and belt tightener/idler assembly. If it is the felt strip being replaced it is necessary to clean away any of the old felt debris and then afix the replacement strip into place on the drum. Note that the felt strip could be installed either on the drum itself or on the round track that is part of the front housing. It is typical for the opposing surface the runs in the track where the felt is placed is lined with a surface made of a hard slippery type of plastic.