Should I tell the university supporting my fellowship application if I apply for other jobs? I am a postdoc currently looking for a faculty level position. I have recently started preparing an application for a grant scheme in the UK. If the application were successful, the grant would pay my salary for 5 years and give me some money to start up a research group. I have contacted a University (let's call them University A), to ask if they would be willing to be the "host institution" for my application. They have agreed, and have been very supportive of my application in terms of giving me help and feedback in preparing the proposal.

Yesterday, I noticed a job advert for another UK University (University B), who are advertising for a lecturer position, specifically seeking someone in my field. I am considering applying for this position.

If I do apply to University B, should I tell University A that I am doing so? In particular, if I were to get the job at B and therefore have to back out of the grant application, having not told A about my application beforehand, will I be seen to have acted in bad faith?

I am in two minds about this. One the one hand, if I was straightforwardly applying for an open position at both A and B, I certainly wouldn't bother informing each of my application to the other. On the other hand, A is investing some non-trivial effort in helping me prepare the grant proposal  so perhaps I owe them more consideration.

A subsidiary question: if I do tell A that I am applying to B, is this likely to dampen their enthusiasm for supporting my grant application?
No.  You are a postdoc.  Everybody knows postdocs need to apply for multiple jobs because the success rate is low.  There is no need to say what other jobs you are seeking.  Treat your fellowship application like a job application.