If I create a new role and assign it "Browse Project" permissions, do other roles lose 'Browse' access? Currently Browse is available to any user.

[Bruce Fint]

Better questions is - where 'any user' is granted permission, how do deny access to this new role (group)?

[Nic Brough -Adaptavist-]

No, the new role is independent of other roles, so adding a new one and granting it browse does not affect the others.

You can not "deny" access.  You have to remove the access that is being granted to the people who you don't want to see it.  In other words, remove "any user" from the permission, and add back in the smaller group(s) of people who need to see it.