I know that was the 'filter' on earlier Savinellis. They then switched to the balsa filters. Smoke with or without, your choice. You may try it both ways and see how you like it. I personally remove all filters from my 'nellis.
Dunhill used the inner tube to prevent the smoke canal in the shank from getting smaller over time. Some smokers weren't keen on keeping their pipes clean so instead of cleaning out the shank or indeed the inner tube they simply replaced it when it was gunked up!
I personally would take it out . It doesn't really cool the smoke or has other real benefits...... apart from being another item that needs cleaning
When aluminum was the rage , Dunhill put a long inner tube in one of their lines [ it has it's own pat number on the tube .. lol ] ... it was said that the tube would cool the smoke . Do they work ? Meh . My Dunhill Innertube smokes fantastic with or without the innertube
I haven't seen an innertube in a Savinelli in many years, and in truth I'd almost forgotten about them until I saw your pic. My Oscar Elite Canadian came with one of those, though I didn't know that it was removeable at the time I got it, about twenty years ago. The original stem got broken a couple of years after I got it, so I had my local pipe shop send it to Lee Von Erck for a new stem. The pipe is still a great smoker, even without the innertube, BTW.
Incidentally, the end of the tube didn't come close to reaching the bowl on that pipe, so I'm really not sure how it was supposed to function.