Never Deadlift Heavy In Running Shoes - Common Strong

"Plenty of people go to the gym trusty, well-worn, comfortable in their running shoes"

But in fact: "These shoes are not made for lifting heavy weights"
Here is the break down of the risks by lifting heavy weights in improper shoes by CrossFit coaches Lee Boyce-C.P.T and Dan McCarthy-an L3

Problems with heavy weight lifting in running shoes:
Running shoes holds their role in supporting the repetitive movement of running, but when it comes to weightlifting it requires a different form of base support, says Boyce. A rigid sole is much better so that the heels can drive into the floor during most workouts such as squats and deadlifts

Alternative: Barefoot lifting?
Some guys in the gym lifting sans shoes, whether in socks or totally barefoot. The barefoot lifting gives you the direct connection to the floor that lifting shoes aim to create.

I doesn't have an issue with barefot lifting during powerlifting movements like squat, bench and deadlift, But without shoes if you are catching a few hundred pounds on your shoulders then your feet would bear the impact and could potentially get injured says McCarthy

The bottom line: Weightlifting shoes are called weightlifting shoes for a reason, They are specialized to help you perform your best and keep your feet from sustaining injuries.