Senior Housing Improvement and Retirement Accounts Act of 2020 or the Senior Housing IRA Act of 2020 This bill amends the Internal Revenue Code, with respect to the tax treatment of gains from the sale or exchange of a principal residence, to (1) allow a qualified individual to contribute the gains from the sale or exchange to a Roth Individual Retirement Arrangement (IRA) as a qualified rollover contribution which is exempt from contribution limits, and (2) increase the $250,000 limit on the exclusion from gross income for gains from the sale of a principal residence by the amount of the rollover contribution. The bill applies to individuals who (1) have attained the age of 55 before the date of the sale or exchange, (2) have owned and used the property as a principal residence for at least 20 years, and (3) have not previously elected to treat a contribution to a Roth IRA as a qualified rollover contribution under the authority provided by this bill.