Going to be just like buffed Heal. AD carry will just take Flash/Barrier. Bot is going to be a bit boring :/
Barrier vs Ignite: The barrier champion wins unless the ignite tick plus any damage dealt is sufficient, ie. your opponent is ahead in terms of overall health/DPS. This means you'll either be sure you're ahead, or save it while escaping once out of the enemy's effective range, to ensure you don't die.
Barrier vs Heal: Number dependent, if we use the live heal comparison I suspect heal will win. Note that heal can be used to negate a large disadvantage in health if you stick to strictly poking, while barrier requires specific timing and will lose out on it's benefit.
Barrier vs Cleanse: Barrier wins unless you're moderately behind, in which case you won't be committing to any fights and it will save your life again and again.
Barrier vs Exhaust: Exhaust wins. You can't afford to commit even if you're moderately ahead in form or health. It really takes a critical advantage to kill someone when you're exhausted. In bot if your support does a fair amount of damage as well, then you might be able to make it out alright, but they're likely going to get the kills over you. Your advantage is exhaust's naturally low range. Poke on, good sir.
Barrier vs Barrier: Static. Other lane factors influence how this will turn out instead.