Lighting – Safe to put insulation around recessed light cans after convert to LED

Yes to causing heat problems, No to using a closed cover.

There is a 3" distance spec around such covers (to the sides), but the top must remain open for non-ic fixtures (to vent their heat).

Air Tight recessed lights might prevent air seepage from a conditioned area to a non-conditioned area, but they can't do much about thermal transfer.

Proper insulation is the only way to fix large amount of thermal leakage. However, your predicament is going to be that your fixtures are not insulation-contact (IC) rated.

With non-IC fixtures, you'll not only find

Do not install insulation within 3" of fixture

But you'll likely also find

Nor above fixture in such a manner to entrap heat

Example

You'll need to check the warning labels and/or instructions for your particular lights, but this all but eliminates the option of putting some sort of heat-trapping box above a fixture with such warning.

You will find it difficult to find a 4" IC rated remodel fixture. They just aren't readily available on the market, except for some speciality LED-driven models.