Do I have to keep my MT4 and NNFX Algo Trader open at all times to trade?

Yes, if you intend for the NNFX Algo Trader to trade automatically without any intervention.
In that case, your MT4 and NNFX Algo Trader must be open during your trading period. To avoid keeping your computer connected 24 hours a day, you can use VPS service (Virtual Private Server).

However, to use the NNFX Algo Trader it is not mandatory to keep the MT4 or the NNFX Algo Trader open all the time. The NNFX Algo Trader can be used just to confirm your algorithm and to give you a hand during Live Trading.
To trade, just open the NNFX Algo Trader, the software will check your algorithm and tell you if there is any action to take according to your rules. You can repeat the process for all pairs and close the MT4 after that.