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Dr. Az-Zalmi, in his book التبيان لرفع غموض النسخ في القرآن, pages 399-400, reports the claim that

was abrogated by the next verse,

His argument, however, is a bit shaky. He says that the job of prophets is to do what 51:55 commands: remind people, then if the reminder has not had an effect, they are not blamed if they turn away from the deniers, which is what 51:54 says to do. Az-Zalmi says that the order to the Prophet is to do what the latter verse, 51:55, says then the former, 51:54.

My humble understanding is that the subject of 51:54 is the disbelievers, while the subject of 51:55 is the believers. That, by itself refutes the abrogation claim. What 51:54 says is that the Prophet (PBUH) may let go of trying to persuade the disbelievers. God won't blame him after he has conveyed the message to them. The true beneficiaries of the message are the believers and they always benefit when they are reminded of it. That's because believers are human and do forget and do sin, but when they are reminded of God's message they remember and repent. That's what 51:55 is talking about.

Al-Jabri says, in his book لا نسخ في القرآن...لماذا؟, page 137, that it was Ibn Hazm Al-Andalusi who made this claim.

For:
Ibn Hazm Al-Andalusi,
Ibn Salaama.

Against:
Al-Jabri,
Dr. Mustafa Zayd,
Dr. Az-Zalmi.

I agree with you, and I would like to point out that 51:55 opens with و which means and, so if there is abrogation, 51:55 has to abrogate itself, too.

Excellent point and I think that's what Al-Jabri was saying in his refutation of this claim on pages 137-138 of his book لا نسخ في القرآن...لماذا؟ He writes that the two verses give two instructions for two different circumstances:

1. When the listeners are closed-minded, turn away from them; it's not your fault.
2. When you see the chance, go back to reminding people, because those who would believe will benefit from repeated reminders.

I like that. Al-Jabri mentions this in his chapter about guidance and education. He writes that people need to have space to digest what they have been told. Those who block their minds will never get it, but those who open them, their hearts may open too.