I had a fire in my apartment on July 27. I didn't have renters insurance. When I applied for the Fire Marshall report it stated that it was inconclusive (they could not determine who's fault it was), now my question is can I sue the management company for compensation of lost items, i.e furniture

Unlikely. The only way you could possibly have a case against them is if you can prove that the fire was caused by a condition known to the management but not corrected. Like faulty wiring or flammable materials stored unsafely. Since, you say the Fire Marshal's report was inconclusive you don't really have a case.

Now go and get rental insurance.

How can I prove that?

Statements from an electrician that the wiring was faulty, pictures that show fire hazards. What do you think?

But, you stated the Fire Marshal's report does not put any blame on them. So where do you think the evidence is going to come from. You are pretty much up the proverbial creek.