Note You may begin the second day of any Jewish festival as early as plag ha'mincha. You should ideally say mincha before plag and then say ma'ariv after plag; but if you are praying with a minyan, you may say mincha anytime after plag and then say ma'ariv immediately afterward, just as on Shabbat.
Note However, there is nothing gained by saying mincha early before a Jewish festival:

• We don't want to start Rosh Hashana (Yom HaDin) early and no one wants to start Yom Kippur early.
• You may not start either seder until after dark on Passover.
• You could eat in a sukka before dark on either of the first two days of Sukkot, but you would not fulfill the requirement of eating in a sukka since it was not dark.  
• Likewise, the first day of Shavuot does not begin until after dark.
• Since you may not say the blessing on eating in a sukka on Shmini Atzeret (which you would have to do if you eat before dark), there is nothing gained by saying mincha and ma'ariv early on that day, either.

So, as a practical matter, the only days on which saying mincha and ma'ariv early would allow beginning the holiday early are the second day of Shavuot, the second day of Rosh Hashana, and the last days of Passover.