It doesn’t say anything specifically about alliances, but it does include references to basing relations between the parties on the principle of sovereign equality, and a commitment from both parties to refrain from the threat or use of force against the other’s political independence - as well as a commitment to work for “mutual partnership and cooperation” with all countries.
The full text can be found here in Finnish and here in Russian - both translations are equally authentic, according to the treaty.
In English, the relevant parts seem to be Articles 1 & 4 - translation taken from here:
Relations between the Parties shall, in accordance with the Charter of
the United Nations and the Final Act of CSCE, be based on such
principles of international law as sovereign equality, abstention from
the threat or use of force, the inviolability of borders, territorial
integrity, the peaceful settlement of disputes, noninterference in
internal affairs, respect for human rights and fundamental freedoms,
as well as equal rights and the right of peoples to
self-determination. The Parties shall conscientiously fulfil their
obligations under international law and, in a spirit of
neighbourliness, work for mutual partnership and coöperation with all
The Parties shall refrain from the threat or use of force against the territorial integrity or political independence of the other Party and shall resolve disputes between themselves by peaceful means in accordance with the Charter of the United Nations and the Final Act of CSCE, as well as other CSCE instruments.
The Parties shall not use or allow the use of their territory for armed aggression against the other Party.
In the event that Finland or Russia is the victim of armed aggression, the other Party shall seek to bring about a settlement of the conflict in accordance with the principles and provisions of the Charter of the United Nations and CSCE instruments, and shall refrain from giving military assistance to the aggressor.