After deciding that the only way to defeat Thanos was to reunite the Avengers, Tony Stark takes out his phone to call Captain America. The only problem is, why is Tony - one of the most technological advanced people on Earth - using an old flip phone, you would think he would just use J.A.R.V.I.S. or some other super-cool techie thing?
Tony Stark uses the flip-phone he received from Captain America at the end of Captain America: Civil War. As the oft-reliable Marvel Cinematic Universe wiki states:
Stark found a phone and letter from Steve Rogers, apologizing for keeping the truth about Stark's parents from him, telling Stark if he would need Rogers' help, he and the rest of his team were just a phone call away.
The scene in Infinity War in which Stark uses the phone is when he is prompted by Bruce Banner to call Steve Rogers to assist them in the fight against Thanos (although he never does end up using it).