Not yet. The NATO treaty provides:
The Parties agree that an armed attack against one or more of them in Europe or North America shall be considered an attack against them all and consequently they agree that, if such an armed attack occurs … will assist the Party or Parties so attacked by taking forthwith, individually and in concert with the other Parties, such action as it deems necessary, including the use of armed force, to restore and maintain the security of the North Atlantic area.
Article 5. But what is an “armed attack”? The protocol admitting Turkey to the pact says:
For the purpose of Article 5, an armed attack on one or more of the Parties is deemed to include an armed attack …  on the territory of Turkey, …  on the forces, vessels, or aircraft of any of the Parties, when in or over these territories…
But neither the protocol nor the treaty define “armed attack.” The Russian warplane did not not fire on Turkish forces but it did violate Turkish airspace and it did so despite many earlier protests by Turkey at similar incursions. The question is then whether the violation of airspace is an “armed attack.”
Like nearly every term in international law, the term “armed attack” has no agreed-upon definition. Not every use of force constitutes an armed attack. As the International Court of Justice has noted, “As regards certain particular aspects of the principle in question, it will be necessary to distinguish the most grave forms of the use of force (those constituting an armed attack) from other less grave forms.” Yet it continues “Every State has the duty to refrain from the threat or use of force to violate the existing international boundaries of another State.” Isn’t sending a military jet a “use of force” that violates Turkey’s boundaries? On the other hand, the jet did not drop bombs on Turkey or try to, as far as we know. For this reason, I suspect that nearly everyone would regard the incursion as a violation of international law but not as an armed attack per se. Meanwhile, Turkey’s response would be deemed a lawful countermeasure and not itself an act of war.
So we are not at war with Russia, but what if Russia retaliates by bombing Turkey? Article 5 supplies wiggle room. Each of the other NATO members is required only to take “such action as it deems necessary” which may, but need not, include the use of armed force. That is the law. Politically, the story may be different.