Can I damage a razor's finish by using Flitz?

I assumed you were talking about polishing a straight razor blade.

If you are talking about polishing a solid brass, bronze or stainless DE razor, you have little to worry about.

If you are polishing a older DE razor that is chrome plating over top of either brass or ZAMAC (if the razor is newer), you have to be careful not to polish right through the chrome plating. If there are pits that already penetrate the plating and you try to polish them out, you will damage the finish.

Just go slowly.

Thanks. And thanks for the warning about the contaminated micro cloth. But what contaminates a micro cloth? Plus you and RayClem refer to not damaging the  edge. What exactly is the edge? The edge of the safety bar? The edge of the top cap?

The edge of a straight razor. Of course it can also dull a DE blade if it gets in contact with metal polish. For a DE you should be fine as long as the cloth you are using is clean and has no dust, dirt or other debris/particles stuck as that can scratch your razor. Even without polish your cloth or towel should always be super clean.

Oh my gosh. I forgot about straight edges. Thanks a lot.

Referring to DEs, or Straights with a gold wash, or possibly an etched blade ... and 'Damascus'. Polishing a gold plated DE razor should only be done AFAIK, with Windex. Any metal polish previously mentioned will take the gold wash off. That also goes for gold on the belly of Straights. Damascus finishes can also be damaged with metal polish, since they are an acid etch to bring out the differing grains in the forged metals. 

Old Gillettes with gold plating have a lacquer over the gold and that's gone in a pair of seconds if you're not careful. Windex is the best bet. As previously mentioned chrome, or rhodium can be polished through to brass if you're not careful. Silver plating is even more delicate. Nickel is pretty tough, but any of them will 'brass' if worn too severely.

Referring to DEs, or Straights with a gold wash, or possibly an etched blade ... and 'Damascus'. Polishing a gold plated DE razor should only be done AFAIK, with Windex. Any metal polish previously mentioned will take the gold wash off. That also goes for gold on the belly of Straights. Damascus finishes can also be damaged with metal polish, since they are an acid etch to bring out the differing grains in the forged metals. 

Old Gillettes with gold plating have a lacquer over the gold and that's gone in a pair of seconds if you're not careful. Windex is the best bet. As previously mentioned chrome, or rhodium can be polished through to brass if you're not careful. Silver plating is even more delicate. Nickel is pretty tough, but any of them will 'brass' if worn too severely.

Good to know. Thank you.

The Flitz package says it's good for brass. But would something like Brasso be better?