If the total fixed costs increase, does this change average (total) costs? How about average variable costs?

Yes, it changes the average (total) costs but the average variable costs remain the same.

Explanation:

Average Total Costs = Total Cost / Total Output

= (Fixed costs + Variable costs) / Total Output

Average Variable Costs = Total Variable Costs / Total Output

Since the total fixed costs increase there will be an increase in total cost, which will ultimately increase the average total costs. Since there is an increase in total fixed costs or fixed costs for that matter, there will be no change in the total output produced as fixed costs have to be incurred always, irrespective of the level of output produced.

Since, there is a change in fixed costs, there will be no change in variable costs and thus there will be no change in average variable costs also.