If you sell your primary residence, invest that gain in a house that you then sell 10 years later for no gain or loss, do you have to pay taxes on the proceeds of sale?

The requirement to defer capital gains by purchasing a residence that was greater than the sale price of the prior home was removed from the tax code in 1997.

If you sold your primary personal residence and you lived in and owned the home for at least two years in the five year period on the date of sale, you do not have to report the sale if your gains are less then the exclusion amounts of $250,000 if filing Single or $500,000 if filing Married Filing Jointly (and both lived in and owned the home for two years). 

Do you have pay capital gains tax on sale of your primary residence if you are older than 75?

Age of the seller is not relevant on whether you pay capital gains or not.

Somebody told my 76 year old dad that- it didn't sound right to me. Thanks!

Maybe some state provision?

Congress eliminated the age 55-and-over capital gains exemption on home sales a long time ago.

Before the $250,000/$500,000 exclusion (before 1997), there was a provision for people over age 55 to exclude up to $125,000 of a home sale.   However, that is long gone.

Although most States follow the Federal rules in this regard, it is possible that your State has different rules.