No. Wait until the required assembly is present, undamaged and correctly fitted. A temporary cloth, mesh patch or omitted filter is not an equivalent substitute.
Understanding this situation
The filter helps manage fibres and the intended air route. A machine that starts without it has not thereby approved that configuration, and a short programme still involves the same concern.
A useful next step
Use another appropriate drying method for the laundry. Confirm the replacement using the complete model and fit it according to the instructions. If another component was damaged when the filter failed, have that assessed before restarting.
Manuals and supporting guidance
Model-specific instructions take priority. These references support the practical principles discussed here; they do not diagnose your appliance.