Two things are going wrong here.
Firstly, your first snippet is not doing what I think you think it is. Try removing the second line, the echo. It still prints the date, right? Because this:
DATE= date +'20%y-%m-%d'
Is not a variable assignment – it’s an invocation of date with an auxiliary environment variable (the general syntax is VAR_NAME=VAR_VALUE COMMAND). You mean this:
DATE=$(date +'20%y-%m-%d')
Your second snippet will still fail, but differently. Again, you’re using the invoke-with-environment syntax instead of assignment. You mean:
# note the lack of a space after the equals sign
FILE="~/path/to/_posts/$DATE-$1.markdown"
I think that should do the trick.
Disclaimer
While I know bash very well, I only started using zsh recently; there may be zshisms at work here that I’m not aware of.