Is it true that in the long run fixed costs are always zero?

Not really. Take a manufacturing example where fixed costs are the cost of plant and machinery; they depreciate / lose efficiency and need to be replaced over time. Same is true for other industries.

They want you to believe that you take care of the fixed costs in the short run, so in almost any economics undergrad class (acct'ing too) they want you to base off that.

Fixed costs are incurred whether a company operates or not. Expenses that are always there.

The only way fixed cost approaches zero is when the company goes out of business.

Variable costs are expenses that change in proportion to the activity of a business... so that can reach zero every once in a while