Does somebody know if the IRremote library is compatible with Moteino boards?
If not, do you know a cheap solution to build a IR-remote controllable via wireless?
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Your Arduino do not need to be wireless to be able to control it wirelessly. If you have a wireless network at home (which most people do), there is usally some ethernet RJ45 ports as well. So instead you use a very cheap ENC28J60 to connect your arduino to the network. You can then access your Arduino wirelessly through your home router. As long as it is connected to your local network its should work.
As for connecting the IR sender and reciever, you need as said before, one digital pin for recieving IR-signals and one digital pin for sending. Which should be no problem even when adding the ENC28J60. It should also work very well with the Moteino if you choose to go that way.