Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) was asked: What is the ruling on congratulating people on the occasion of the Hijri new year, and how should one reply to a person who offers congratulations? 

He replied: 

If someone offers you congratulations, then respond to him, but do not initiate such greetings. This is the correct view concerning this matter. So if a person says to you, for example, “Happy New Year”, then you can say, “May Allaah make it a good and blessed year for you.” But you should not initiate such a greeting, because I do not know of any report that the salaf [early generations of Islam] congratulated one another on the occasion of the new year, rather the salaf did not regard the first of Muharram as the first day of the new year until the caliphate of ‘Umar ibn al-Khattaab (may Allaah be pleased with him. 

Shaykh ‘Abd al-Kareem al-Khudayr said concerning offering congratulations on the occasion of the hijri new year: 

Praying for another Muslim in general terms, in phrases that are not meant as a kind of ritual  on special occasions such as Eid, is acceptable, especially if what is meant by this greeting is friendship and to show a friendly face to one’s fellow Muslim. Imaam Ahmad (may Allaah have mercy on him) said: “I do not initiate the greeting but if someone greets me I return the greeting, because responding to the greeting is obligatory. But being the first to offer congratulations is neither Sunnah nor forbidden.