The TCP/IP stack and wifi modem rule out most microcontrollers, they don't have either the memory or speed to deal with IP networking and wifi adapters. Handling that requires a lot of software, that just isn't going to fit in the 32k that microcontrollers typically have to work with.
That said, anything complex enough to handle the wifi and IP stuff will work just fine for driving an IR LED.
The ESP8266 would work fine for what you want. It doesn't need to be programmed in LUA with nodeMCU, you can program the things with the Arduino software as well. You'll need a USB-TTL cable to program it, which will need Rx/Tx/DTR broken out. The downside of the ESP8266 is that it needs a lot of power, at least a few hundred mA. You'll also be writing a lot of code, as there probably aren't Arduino libraries for the JSON parsing you'll need.
A Raspberry Pi (or one of the clones) is a good choice too. It won't use that much more power than the ESP8266, and has a pin header for the GPIO you need for the IR, and has an entire Linux distro running on it. A big plus, as the software you need for parsing JSON is already there, and the networking is standard Linux stuff (well documented and works well), and you can use the Linux LIRC package to handle all of the IR stuff. I would do this if you aren't making thousands of these things. Even then, I would do this first as a prototype.