On current iPhones, ‘Sleep Mode’ can mean two different things: Sleep Focus, which limits interruptions, or the display going to sleep through Auto-Lock. Use the section that matches what you want to turn off.
Turn Off Sleep Focus
Open Control Center, tap Focus, and turn off Sleep when it is active. If Sleep keeps returning automatically, review the Sleep Focus schedule and related Sleep settings rather than repeatedly turning it off manually.
Change the Sleep Schedule
Sleep schedules are managed with Apple’s sleep features, including the Health app’s Sleep section and related Focus scheduling. Adjust the schedule if Sleep Focus is activating at unwanted times.
Keep the Screen from Sleeping
If you mean the display turning off, open Settings > Display & Brightness > Auto-Lock and choose a longer interval or Never when that option is available. Low Power Mode or device-management policies can affect available choices.
Do Not Confuse the Two Features
Sleep Focus controls notifications and Focus behavior. Auto-Lock controls how quickly the display turns off. Changing one does not automatically change the other.
Frequently Asked Questions
Why does Sleep turn on every night?
A Sleep schedule or automation may be enabling Sleep Focus.
Why can’t I choose Never for Auto-Lock?
Low Power Mode or device-management restrictions can affect Auto-Lock choices.
Will turning off Sleep Focus change my alarm?
Focus and alarms have separate behavior; review your alarm/sleep schedule before making schedule changes.
Summary
Turn off Sleep Focus from Control Center if you want notifications/Focus behavior restored. Change Auto-Lock if your goal is to keep the screen awake.

Matthew Roberts created this site to help his friends and family with questions that they might have about how to change the settings on their iPhones. It now serves as an archive of iPhone how-to guides to help anyone that stumbles upon this website.