Finland is not very well connected to the rest of Europe by train. Finland shares borders with three countries: Russia, Norway, and Sweden, and of those, there only exists a cross-border passenger train service to and from Russia. So, if you're determined to make the entire journey by train, then you'll have to go through Russia and pay for a visa.
Here's one route which looks pretty direct to me. You may need to change trains to complete some of these journeys.
Rome to Vienna (Austria)
Vienna to Breclav (Czech Republic)
Breclav to Warsaw (Poland)
Warsaw to Minsk (Belarus)
Minsk to St. Petersburg (Russia)
St. Petersburg to Helsinki (Finland)
If you're willing to include one ferry (which I think is included in Interrail tickets anyway), then you could take this faster, shorter route:
Rome to Innsbruck (Austria)
Innsbruck to Munich (Germany)
Munich to Copenhagen (Denmark)
Copenhagen to Stockholm (Sweden)
Stockholm to Helsinki (Finland) by ferry.
Or just fly