Short answer
Not as a festival. The Islamic New Year (1 Muharram) has no special prayer, feast or sunnah celebration in the Hanafi school or the other three. Muharram is a sacred month, though, and fasting in it, especially on Ashura, carries great reward. A simple dua for a blessed year is fine.
There is no act of worship specific to 1 Muharram, so treating the day as a religious festival with set rituals is not part of the deen. Fasting in Muharram is mustahabb (recommended), and fasting Ashura on the 10th together with the 9th or 11th is sunnah. Exchanging a general dua for a good year is permissible as long as it is not treated as a sunnah.
The four schools agree on this.
Explanation
The Islamic calendar counts its years from the Hijrah, the Prophet's ﷺ migration from Makkah to Madinah. In the time of Umar ibn al-Khattab the Companions agreed to date the years from that event, and Muharram was kept as the first month. Neither the Prophet ﷺ nor the Companions marked the first day of the year with a celebration, and the four schools agree there is no special rite for it.
What the month does have is virtue. Muharram is one of the four sacred months the Quran mentions, and the Prophet ﷺ called it the month of Allah and said the best fasting after Ramadan is fasting in it. Its best known day is Ashura, the 10th. The Hanafi scholars recommend fasting the 9th and 10th, or the 10th and 11th. See Ashura and fasting only the 10th.
- Avoid: parties, fireworks or countdowns copied from the Gregorian New Year, and any claim that a special prayer or dua for the last night of the year is sunnah. The messages that circulate with such duas have no reliable source in the hadith collections.
- Fine: saying to someone "May Allah make this year blessed for you". It is a good dua. Just do not make it a required custom or a religious greeting.
- Good habits: take stock of the year that passed, repent, plan to fast in Muharram, and learn about the Hijrah.
For the date and countdown, see Islamic New Year and the Islamic calendar. For the Gregorian New Year, see celebrating New Year's.
The evidence
Quran, Surah At-Tawba 9:36
إِنَّ عِدَّةَ ٱلشُّهُورِ عِندَ ٱللَّهِ ٱثْنَا عَشَرَ شَهْرًۭا فِى كِتَٰبِ ٱللَّهِ يَوْمَ خَلَقَ ٱلسَّمَٰوَٰتِ وَٱلْأَرْضَ مِنْهَآ أَرْبَعَةٌ حُرُمٌۭ ۚ ذَٰلِكَ ٱلدِّينُ ٱلْقَيِّمُ ۚ فَلَا تَظْلِمُوا۟ فِيهِنَّ أَنفُسَكُمْ ۚ وَقَٰتِلُوا۟ ٱلْمُشْرِكِينَ كَآفَّةًۭ كَمَا يُقَٰتِلُونَكُمْ كَآفَّةًۭ ۚ وَٱعْلَمُوٓا۟ أَنَّ ٱللَّهَ مَعَ ٱلْمُتَّقِينَ ﴿٣٦﴾
Indeed, the number of months with Allah is twelve [lunar] months in the register of Allah [from] the day He created the heavens and the earth; of these, four are sacred. That is the correct religion, so do not wrong yourselves during them. And fight against the disbelievers collectively as they fight against you collectively. And know that Allah is with the righteous [who fear Him].
The year has twelve lunar months, four of them sacred, and Muharram is one of the four.
Sahih Muslim 1163
Sahih... أفضل الصيام بعد رمضان شهر الله المحرم ...
"The best fasting after Ramadan is in Allah's month of Muharram."
The virtue of Muharram lies in fasting, not in a new year celebration.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Sawm, the discussion of recommended and disliked fasts. Lists fasting Ashura with a day before or after among recommended fasts.
- Nur al-Idah with Maraqi al-Falah, Kitab al-Sawm, the section on the kinds of fasting. Counts fasting in the sacred months and on Ashura as recommended.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.