I’m afraid that there is no clear or agreed answer to your question. Our opinion is that hot drinks will not be helpful to cold sores.
1. If you have an active cold sore, heat will tend to irritate the area, increasing inflammation.
2. We also know that sunlight can reactivate cold sores and this is the reason that some people are more affected by these during the summer months. This is probably because UV light activates the viruses, but heat could also play a part.
In the end, we know that cold sore viruses are peculiar things and that what is a significant factor in one person may not be relevant for the next.