Straight answer first, because this query deserves one: no iPhone app can control a TV with no network connection whatsoever. iPhones have no infrared hardware (never have), and TVs don’t accept remote input from phones over Bluetooth. Any app promising IR-style control from an iPhone is promising physics it doesn’t have.
But hold on, because “I don’t have Wi-Fi” almost never means “zero network is possible,” and the actual situations behind this search all have answers.
Does a TV remote app need internet, or just a network?
Every command a remote app sends is local traffic: phone to TV, never touching the internet. A network with no internet behind it works completely. That unlocks three situations:
No home internet / router. Your iPhone’s Personal Hotspot is a network. Join the TV to the hotspot (TV settings → network → your phone’s name), and the remote app on that same phone controls the TV. Cell signal irrelevant; the hotspot works in airplane-mode-with-hotspot-on conditions.
Internet is down. Wi-Fi router still powered means the network still exists. The app keeps working during an outage; only the streaming suffers.
The TV lost its network and the remote is also gone. The chicken-and-egg case, and the hotspot has a trick for it: TVs automatically rejoin networks they remember. Rename your iPhone (Settings → General → About → Name) to the old network’s exact name, set the hotspot password to the old network’s password, turn the hotspot on, and the TV walks right in, where A Decent Remote on the same phone takes over. Full walkthrough with brand specifics in the Roku and Fire TV guides.
When is phone control genuinely impossible?
A non-smart TV with no network hardware (the garage CRT, the 2010 plasma) cannot be reached by any phone app. The honest fix is a $10 universal IR replacement remote from any big-box store; they cover decades of IR codes for every major brand. (This is also the fallback for 2019+ Panasonic sets, which are network-locked.)
For everything made in the smart-TV era, though, the phone in your pocket covers it, and the network it needs is one you can create yourself. No internet required.