Well, no, installing and using the Azure Boards GitHub App does not by itself consume GitHub Actions minutes or provision GitHub Actions runners.
The Azure Boards app is a GitHub App used for repository integration and traceability between GitHub artifacts and Azure Boards work items.
But if an action performed through the integration generates a GitHub event such as a commit or pull request, and your repository contains a GitHub Actions workflow configured to trigger on that event, that workflow can run normally. Aa any Actions usage would then be caused and billed as a GitHub Actions workflow run, not by the Azure Boards integration itself.
And so to assess the impact on Actions usage, is better to check the repository's .github/workflows/*.yml trigger definitions (on:), rather than your Azure Boards App installation itself.
https://learn.microsoft.com/en-us/azure/devops/boards/github/?view=azure-devops
https://learn.microsoft.com/en-us/azure/devops/boards/github/install-github-app?view=azure-devops