I have buildings with icons above them. Some I can tap and it'll go away but some stay and cannot access..

This will depend on the building (and maybe the icon shown). Make sure the building is connected by road to the base otherwise it wont be active.

Some buildings like the VIP shop need to be unlocked with money and will have a padlock icon above it until you do that. The military intel building will show a news paper at you for news reports but it should also let you access some other functions. The clone Lab will only be active for a few days each week, if you have dead troops to clone.

If this doesn't help though please let us know the building that is the problem and describe the icon you are seeing.

Wrench icon above the research academy