How to eliminate vibration noise that started after replacing blower motor?
If the motor uses a belt to connect to the fan it is possible that:


A bearing went bad along with the original motor and now under speed and the air drive load that the new motor taxes that bearing toward the point of destruction. 
The motor pulley is not lined up perfectly with the pulley on the air circulation fan. As a result the V-Belt may be causing a vibration as the belt alternately tries to climb out of the pulleys and then fully engages the pulleys whilst running at speed.
The set screw that attaches the pulley to the motor shaft may not be tight.
The hole in the pulley may be worn such that it is not a secure and smooth fit on the motor shaft.
The belt may be damaged and has a weak spot that twangs each time it goes around at speed.


If the fan rotors directly mount on the motor shaft it is possible:


Sometimes there are two motor shafts, one on each side of the motor. Two separate fan rotors would be mounted on the two shafts. The two rotors in such case may have been originally balanced by adding small clip on weights to the blades of the fans. If they were re-assembled to the new motor with a different rotated offset from one another than the original then the balance weights may now be creating an imbalance.
If the fan rotor is a type that has a single mount to the motor shaft it is common that this mount is to a centered (from side to side) support plate on the rotor assembly. It is possible that the plate is made of cheap thin metal and can be bent just enough that the fan rotor gets a run out variation as it makes each turn. Even a small apparent run out of say an eighth inch or so can create quite a vibration at full speed.