Small part of filler in furnace is stuck. Can I still insert a new filter?
I would say possibly. If you can get the new filter to seal you may be able to install a new one but probably not. 
I say probably not because the piece of the old filter may allow air to bypass the filter. 
When you run the system without the filter all the dust and debris that normally get stuck on the filter end up in the heat exchanger. An excessive amount of dust can even cause a fire on the breathing air side of the heat exchanger. The mammal dust burning is usually not hazardous. The problem is now the ash and other debris stuck in there shorten the life of the fire box. 
If your system has ac coils in the air handler or is a heat pump the dust plugs the coils and quickly reduces the efficiency and in the summer months may cause icing of the coils.

For these reasons it is not a good idea to leave part of a filter that can affect the seal, I have a long 3’ pair of finger grabbers, you push a plunger and 4 metal fingers come out I then grab the part I cannot reach releasing the plunger and pulling the piece out. Many auto parts stores carry these, in some cases I have had to disassemble the duct to repair the channels that hold the filter.