Thank you for this very important question. Many people have written to The Refiner's Fire asking the same thing.
Will people be punished for celebrating the Messiah's birthday on the wrong day? It depends on how they celebrate His birthday, and what they make of it. Scripture never commands us to observe and celebrate His birthday so remembering it and celebrating it is certainly not wrong in-and-of itself.
Most of the church knows that "Jesus" wasn't born on December 25th, so why are they celebrating it then, in the first place? Should we celebrate our own birthdays in a different day and month as well? Does that even make sense? So if the Messiah's birthday is to be celebrated, it seems only right that it would be observed and celebrated on the proper day, and not moved to some arbitrary day in December!
The Scriptures reveal He was born on the Feast of Tabernacles/Sukkot, which falls on our Gregorian calendars in the September/October timeframe.
In reality, though, the problem is not that His birthday should or should not be celebrated, rather, it is the fact that Christianity came established a pagan holiday called Christmas to celebrate it! And what is done at Christmas? People spend huge sums on each other, many going into debt, for the exclusive purpose of giving gifts to each other. When the Messiah was born, did the Magi seek Him, go to Him, then give gifts to each other? And what was the reason the Messiah was born? Was it to create a "new" holiday where people will party, erect trees in their home decorating them with lights and colorful balls, and take the day off work? (Decidedly not! In fact, erecting trees in ones home and decorating it is a prohibited pagan practice - see Jeremiah 10:2-5.) No, the reason the Messiah was born was to become our life substitution one day, so we could have eternal life in the Kingdom provided we straighten up and obey. (John 14:5; 17:2-3).
So you see "Christmas" would not be bad to do, if it actually celebrated the Messiah's birth in a manner that honored Him and YHWH and people spent time learning and talking about what it meant. But "Christmas", as it is celebrated today the world over, is sadly off the mark.
At the same time, Easter is also off the mark, because the Resurrection happened at Pssover (see "The pagan origins of Easter"). It's His death and resurrection that are important - not dressing in one's Sunday finest and watching the sun rise, and having kids gather colored eggs.
So, the answer to your question is not "will we be punished", rather, it should be "does it matter!": Yes, it matters to YHWH. He actually commanded His people to keep His feasts, His annual appointed times, and we really don't have a right to veer from that and make phony holidays more important.
Also, it might have been better to ask: Do you believe God will punish us for celebrating His Son's birthday?" You see, YHWH, God doesn't have a birthday, but His Son (a human with a divine Nature) was born on the first day of Sukkot and circumcised on the eighth day. Yeshua definitely carried YHWH's "qnoma" (see Explaining Yeshua), but His Father is Spirit (Genesis 1:2, John 4:24). Yeshua came from the Father; He was the "arm" of YHWH (Isaiah 53:1).
This is akin to hackneyed phrase used by Catholics who call on "Mary, mother of God." GOD doesn't have a mother; however, the SON did....