Nice piece with nice rhyming. However, I have some critiques:

Why can't winter stay?
If only for a day,
To take away the hot,
But the cold may still remain.

'To take the the hot away' sounds better for rhyming

Why can't winter stay?
Blistering winds that sway-
Trees, that rustle, as if pleading,
To let the cold remain.

'Blistering winds that way
Trees rustling, as if pleading'

Why can't winter stay?
To let the dawn be dusk,
As snowflakes fall like tears of joy,
To let the cold stay just.

'fall' should be 'falling'
'to let the cold just stay'

Why can't winter stay?
The night you stole my heart away;
You held me close, and kissed my nose,
December it should stay.

I think you should remove the semicolons (;)
and the comma in the middle and end of 3rd line
'December it should stay' is not understood in this way unless you remove 'it'.

This aside, I enjoyed reading.