We've seen countless reports and claims from executives and bosses at HBO that all seem to suggest that the plan is to release House of the Dragon's second season sometime in summer 2024. With that plan in place, it would make sense for the second season to start filming very soon, so that there will be plenty of time to work through post-production to meet the summer 2024 window, and surprise, surprise, HBO has announced just this.
As House of the Dragon Season 2 is officially in production. This means that the show is being filmed as of now, and will likely continue to do so for a couple of months at least.
<social>https://twitter.com/hbomax/status/1645804061437095939</social>
As for the actual release date of the series, HBO has not expanded on this any further.