Hi F,
Restarting Diagnostics & Health Monitoring Services shouldn't directly impact your production client services.
Said that, please note these are the services which monitors and fixes issues automatically for you. Hence any service goes down, you are on your own to figure out and fix them.
So best recommendation would be put your server on DAG maintainance mode
one at a time. Then proceed with the change.
Why do I need
Exchange 2013 Maintenance mode?
This is to ensure that your end users wont be effected when performing any type of software/hardware maintenance on your exchange servers or when you are working on troubleshooting Exchange related problems.
DAG members will have extra steps involves moving all active databases off the server and blocking active databases from moving to the server. Also we need to ensure that all critical DAG support functionality that may be on the server (for example, the
Primary Active Manager (PAM) role) is moved to another server and blocked from moving back to the server.
References:
Disabling MSExchangeHM service
Performing maintenance on DAG members