If I sell shares in a mutual fund and reinvest the proceeds in another fund from the same fund family, do I have to pay tax on any gain?

Yes, since you are actually selling one fund and purchasing a new fund. You need to report the sale of the shares you sold on Form 8949, Sales and Dispositions of Capital Assets. Information you report on this form gets posted to Form 1040 Schedule D. You are liable for Capital Gains Tax on any profit from the sale. The fact that you reinvested the proceeds in another mutual fund has no bearing on your tax liability for this sale.