The Hashiras would likely show concern. Some, like Shinobu, would immediately start looking into possible medical solutions. Giyuu might be more reserved in his worry but would still keep a close eye on Tanjiro. The other demon slayers would also be affected. They look up to Tanjiro, so his sickness would lower the morale of the group. They would try to support each other while waiting for Tanjiro to recover.
There's a fanfic called 'Gabimaru and Sagiri: The Unlikely Alliance'. In this one, they start off as reluctant partners but gradually form a deep connection. The author does an amazing job of showing Gabimaru's protective side towards Sagiri and how Sagiri comes to respect Gabimaru's skills and determination. It has some really intense and emotional scenes that make it a great read.
Tanjiro might use his sense of smell to detect hidden threats. Since his sense of smell is super - enhanced, he could sniff out any demons or traps in the way of saving Rengoku. For example, if there are demons disguised as allies, Tanjiro would be able to tell by their scent.