Sometimes a Gmail thread can continue long beyond the point where the conversation ceases to interest you. It can be an annoyance to have these messages cluttering your inbox, but Gmail provides a simple mechanism to eliminate them.
In your Gmail interface, open or select a message from the unwanted conversation, click on More and select Mute:
Any new messages later added to the conversation will bypass your inbox so that the remaining conversation stays archived. If you need to find a muted conversation, or if you accidentally muted a thread, your muted messages are not marked as read, and are still searchable. Simply use is:muted into your Gmail search box to find them.
Muted conversations will re-appear in your inbox if a new message in the conversation is addressed only to you, or if you’re added to the To or Cc line of a new message.
Any relevant filters you have set up will still be applied to muted messages.
To un-mute a conversation, select the message and, once again, select More. You’ll see that the Mute option has now become Unmute, which you can select to un-mute the thread.