Will Shimano Acera 8 speed derailleur work on 6 speed freewheel?

It will work, at least with respect to the crucial functionality of correctly aligning the chain with the frewheel cogs, because all Shimano rear derailleurs intended for between 6 and 9 gears have the same derailleur shift ratio (equal to 1.7), which is the ratio between the sideways movement of the derailleur and the amount the cable is pulled. This means that both the Tourney and the RD-M360 derailleurs will move laterally the same amount for the same command of your shifter, so both will correctly align the chain with the frewheel cogs.