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.
One great 'Muichiro x Reader' fanfiction is 'A Meeting in the Mist'. It beautifully captures Muichiro's somewhat aloof nature while gradually developing a sweet relationship with the reader. The descriptive language used really makes you feel like you're in the Demon Slayer world with them.