No.  You are not responsible for a late-enrollment penalty (LEP) if you  become eligible for the Medicare Part D Extra Help or Low-Income Subsidy (LIS) program.

However, you may still be responsible for your unpaid late-enrollment penalties prior to becoming eligible for Extra Help.  As the Center for Medicare and Medicaid Services (CMS) notes:

"[I]f you owe a late enrollment penalty prior to <effective date of LIS eligibility> you are responsible for paying that amount." 

And remember, you automatically qualify for the Medicare Part D Extra Help program if you qualify for Medicaid (or are dual eligible Medicare/Medicaid).

You can read more about the Medicare Part D Extra Help program here: https://q1medicare.com/PartD-Medicare-Medicaid-Extra-Help-LIS.php

For more information . . .

Please see Chapter 4 of the CMS Medicare Part D manual, Section 10.3.3 - Creditable Coverage Period Determinations for Low-Income Subsidy (LIS) Eligibles:

"Pursuant to 42 CFR 423.46(a) and 42 CFR 423.780(e) Medicare beneficiaries who qualify for the low-income subsidy (LIS) may enroll in a Medicare prescription drug plan with no penalty. Therefore, Part D plan sponsors are not to make creditable coverage period determinations for any new enrollee who is LIS eligible at the time he/she makes the enrollment request or at the time the enrollment becomes effective." [emphasis added] (p.10)

The same Medicare manual then provides in Section 30.4 - Reporting Adjustments to Creditable Coverage Period Determinations Previously Reported to CMS, B. Reporting Adjustments Due to Low-Income Subsidy Eligibility:

"Pursuant to 42 CFR §§ 423.46(a) and 423.780(e), individuals who are LIS-eligible are exempt from being assessed an LEP. Therefore, if a beneficiary currently paying an LEP becomes LIS-eligible, the penalty is removed effective with the start of LIS eligibility." (p.23)

To help with clarification, the Medicare manual also provides the following example using Medicare plan [ABC]:

"Mr. Johnson enrolled in Part D for the first time effective January 1, 2007.  His Part D IEP [initial enrollment period] ended on May 15, 2006, and, because he had no other creditable coverage prior to enrolling in Part D, he had 7 uncovered months, and was charged an LEP [late-enrollment penalty].  He became eligible for LIS [low-income subsidy or Extra Help] effective February 1, 2008, and his plan, Plan [ABC], was notified of this change through CMS plan reports released during the month of February.  Plan [ABC] must submit a plan change transaction (73) with the creditable coverage flag set to “R” and the number of uncovered months value set to “000” with the effective date of February 1, 2008 because this is the effective date of his LIS eligibility." (p.24)

Finally, in the Medicare manual appendix, CMS also provides a sample letter that a Medicare beneficiary will receive letting the person know that they are no longer responsible for paying their late-enrollment premium penalties based on their newly-instated Low-Income Subsidy eligibility status:

"Exhibit 5: Model Letter - Informing Beneficiary of the Removal of the LEP Due to LIS Eligibility (HPMS Code 8010)

<Date>
Dear < Insert Name of Enrollee >:

We are writing to inform you that your monthly premium will no longer include a late enrollment penalty amount that you were being charged.

You will no longer be charged a late enrollment penalty because Medicare’s records show that effective <effective date of LIS eligibility> you were receiving extra help from Medicare to pay for your prescription drug coverage.

This means that your monthly premium will be reduced by <insert amount of LEP>. Therefore, your new premium amount will be <new premium amount>.  [For members in direct-bill status, insert the following language:]

This also means any late enrollment penalty amount that you’ve paid since <effective date of LIS eligibility> [Select method of LEP refund:] will be refunded back to you as soon as possible OR will be applied to reduce your next bill.

We will [Select method of LEP refund:] refund you /reduce your next bill by <total LEP amount since the effective date of LIS eligibility>.   However, if you owe a late enrollment penalty prior to <effective date of LIS eligibility> you are responsible for paying that amount.

[OR insert the following for members in premium-withhold status:]

This also means that any late enrollment penalty amount that you’ve paid since <effective date of LIS eligibility> will be refunded to you by the Social Security Administration. The Social Security Administration will refund you <total LEP amount since the effective date of LIS eligibility> as soon as possible. However, if you owe a late enrollment penalty prior to <effective date of LIS eligibility> you are responsible for paying that amount.

If you have questions about the information in this letter or if you would like more information about the late enrollment penalty, call <Plan Name> at <toll-free number> <days and hours of operation>. TTY users should call <toll-free TTY number>. You can also get information by visiting www.medicare.gov on the web or by calling 1-800-MEDICARE (1-800-633-4227). TTY users should call 1-877-486-2048.

Thank you."

[emphasis added] (pp.64-65)

As a reminder about the late-enrollment penalty . . .

A Medicare beneficiary may incur a late-enrollment penalty if they have a continuous period of 63 days or more after the end of their Part D initial enrollment period (IEP) without some form of creditable prescription drug coverage (such as VA or employer drug coverage).

Or as noted by the Centers for Medicare and Medicaid Services (CMS):

"Under §1860D-13(b) of the Social Security Act, and 42 CFR §423.46, 423.56(g), Medicare beneficiaries may incur a late enrollment penalty (LEP) if there is a continuous period of 63 days or more at any time after the end of the individual’s Part D initial enrollment period during which the individual was eligible to enroll, but was not enrolled in a Medicare Part D plan and was not covered under any creditable prescription drug coverage. “Creditable prescription drug coverage” is coverage that meets Medicare’s minimum standards since it is expected to pay, on average, at least as much as Medicare’s standard prescription drug coverage." (p.5)

Another note about losing your LIS status and the late-enrollment penalty:

If your status should change and you are no longer eligible for the Extra Help or LIS program, you can be subject again for a late-enrollment penalty if you have a period of 63 days without creditable prescription drug coverage.

As note in this same Medicare manual, Section 10.3.3 - Creditable Coverage Period Determinations for Low-Income Subsidy (LIS) Eligibles:

"[S]hould the enrollee lose his/her LIS status, but remain continuously enrolled in a Part D plan sponsor, the Part D plan sponsor shall not make a creditable coverage period determination following such loss for any period prior to their loss of LIS.  [However]. [t]he Part D plan sponsor shall make a creditable coverage period determination, as described in §10.2 [Determining Whether There Has Been a Break in Creditable Prescription Drug Coverage], only if the individual loses his/her LIS-eligibility, disenrolls from a Part D plan sponsor, incurs a qualifying gap in creditable prescription drug coverage, and is not LIS-eligible at the time of reenrollment or at the time the enrollment is effective in a Medicare prescription drug plan." [emphasis added] (p.10)

Additional reference:

MEDICARE PRESCRIPTION DRUG BENEFIT MANUAL, CHAPTER 4 Creditable Coverage Period Determinations and the Late Enrollment Penalty updated: December 21, 2009 (revised: January 5, 2018)