@Naik, Dhara Thanks for reaching out.
Can you share more details on the below:
Function app trigger stops processing the messages and messages are getting lost.
When you mentioned stop processing the message then you mean that the function app is not getting triggered. Have you verified if the message are send/consumed from the entity(queue, subscription) using the portal metrics if there are any outgoing/incoming message to that entity. You can also verify using service bus explorer to validate if the message are still there.
When we send another message immediately it works.
When you send the test message then your function app gets triggered so this confirmed that the message was send successfully and triggered and consumed by your function app.
By the above test it confirms that there is no issue with the function app and it runs successful.
Now you should validate this "Our service bus will receive message from a third party system".
You should validate whether the message is successfully sent by your third-party system/application as there would be issue while sending the message as your test was successful. The same can also be validated by reviewing the incoming message metric for your entity. Once you confirm that there is no incoming message metrics for the timeframe that your third party has send the message then you need to reach out to your third party to validate their application logs to validate if they have send the message successfully.
Update:
I'm glad that you were able to resolve your issue and thank you for posting your solution so that others experiencing the same thing can easily reference this! Since the Microsoft Q&A community has a policy that "The question author cannot accept their own answer. They can only accept answers by others", I'll repost your solution in case you'd like to "Accept" the answer.
Issue:
You observed that Function app trigger stops processing the messages and messages are getting lost. When we send another message immediately it works.
Solution:
You were able to figure out the issue. There was another function that was configured on the same queue and that function was reading the message from the bus and we were not able to see those message.
Thank you again for your time and patience throughout this issue.