No - requests to getLogs do not count towards your usage. They don't count as a search request, so they have no effect on your bill.
They do, however, count as an indexing operation, and indexing operations are subject to a volume limit even on plans that don't bill for them.
The limit that applies
On plans released July 2020 and later, indexing operations aren't billed, but there's a contractual limit of 10,000 indexing operations for every 1,000 search requests and records used per month. Heavy getLogs polling counts towards that limit.
If you hit it you may see an error telling you the indexing limit has been reached. The fix is to reduce how often you call the endpoint, for example, by polling less frequently or narrowing the log types you request.
For the full definition of what counts as an indexing operation, and how the limit works on legacy plans, see How does Algolia count indexing operations?