Does Allah forget according to Quran?

Allah does not forget (Qur'an 20:52):

قَالَ عِلْمُهَا عِندَ رَبِّي فِي كِتَابٍ لَّا يَضِلُّ رَبِّي وَلَا يَنسَى

[Moses] said, "The knowledge thereof is with my Lord in a record. My Lord neither errs nor forgets."

The word نسي has multiple meanings as can be seen in Al-Mo'jam:

1. Forget, fail to remember, fail to recall
2. Fail to care for
3. Leave out or not include
4. Fail to notice or ignore
5. Etc.

Qur'an 7:51, the verse you mentioned, is using the word to mean overlook, rather than forget -- it is a form of muqabalah:

الَّذِينَ اتَّخَذُوا دِينَهُمْ لَهْوًا وَلَعِبًا وَغَرَّتْهُمُ الْحَيَاةُ الدُّنْيَا فَالْيَوْمَ نَنْسَاهُمْ كَمَا نَسُوا لِقَاءَ يَوْمِهِمْ هَذَا وَمَا كَانُوا بِآيَاتِنَا يَجْحَدُونَ

Who took their religion as distraction and amusement and whom the worldly life deluded. So today We will forget them just as they forgot the meeting of this Day of theirs and for having rejected Our verses.

In Tafsir Al-Qurtubi (الجامع لأحكام القرآن - تفسير القرطبي)

فَالْيَوْمَ نَنْسَاهُمْ أي نتركهم في النار كَمَا نَسُوا لِقَاءَ يَوْمِهِمْ هَذَا أي تركوا العمل به وكذبوا به

His saying "So this Day We shall forget them" means they will be left out in the hell fire as they have overlooked the day of judgment, did not work for it, and denied it.

In Tafsir Ibn Kathir (تفسير ابن كثير للقرآن العظيم):

قوله فَالْيَوْمَ نَنْسَاهُمْ كَمَا نَسُوا لِقَاءَ يَوْمِهِمْ هَذَا أي نعاملهم معاملة من نسيهم ... وإنما قال تعالى هذا من باب المقابلة ... قال العوفي عن ابن عباس في قوله فَالْيَوْمَ نَنْسَاهُمْ كَمَا نَسُوا لِقَاءَ يَوْمِهِمْ هَذَا قال: نسيهم الله من الخير ولم ينسهم من الشر

In saying "So this Day We shall forget them as they forgot their meeting of this Day, and as they used to reject Our Ayat," it means Allah will treat them as if they were forgotten, and this is a form of muqabalah. Al-'Awfi said through Ibn-'Abbas that Allah will forget the good about them, but not their evil.

Or you can refer to the abridged version online:

'Ali bin Abi Talhah reported that Ibn 'Abbas said, "We shall forsake them as they have forsaken the meeting of this Day of theirs." Mujahid said, "We shall leave them in the Fire." As-Suddi said, "We shall leave them from any mercy, just as they left any action on behalf of the meeting on this Day of theirs."

In Zahrat Al-Tafasir (زهرة التفاسير), Abu Zahra said:

أريد بالنسيان لازمه وهو الترك بل بعض علماء اللغة يقول: إن الأصل في معنى النسيان هو الترك

What is meant by forget is to leave out, as linguistic scholars say the origin of the word نسيان is to leave out [or overlook].

Muqabalah is a known linguistic technique in the Arabic language (and other langauges), which is a type of tibaq, where the same word can be used to convey different meanings. There are several verses in the Qur'an that employ this linguistic technique, e.g., Qur'an 30:55, where the word ساعة is used to mean hour (the commonly understood meaning) as a figure of speech to denote that the criminals' lives were so short, and also to refer to the day of judgment:

وَيَوْمَ تَقُومُ السَّاعَةُ يُقْسِمُ الْمُجْرِمُونَ مَا لَبِثُوا غَيْرَ سَاعَةٍ كَذَلِكَ كَانُوا يُؤْفَكُونَ

And the day the hour appears the criminals will swear they had remained but an hour. Thus they were deluded.