standups and voice-overs
Read through the script several times to get familiar with it and get over any bumps in your delivery during practice - not while you're recording.
Look especially for words that are going to be hard to pronounce or that you might stumble over. Repeat those words out loud to yourself until you feel comfortable with the pronunciation. Or try doing exaggerated pronunciations of a word several times to get more at ease with just saying it naturally.
Do some facial stretches - opening your mouth wide to loosen up.
Try humming or singing a song to get your voice alive and prepare it for using a range of pitches when you narrate your story.
Say some classic tongue twisters out loud - like "Peter Piper picked a peck of pickled peppers" - and emphasize each of those words.
To loosen up and sound more conversational, try laughing through the copy. It won't make any sense but it will warm you up and get you to be yourself.
Go to extremes when practicing your script to compensate for problems you have with your voice. Thus you might try reading the script in a very loud voice if your problem is having too soft a voice.
For standups, relax your nerves by shaking your hands from the wrists. Also do the facial stretches and take some deep breaths.
Do deep breathing exercises - breathing in quickly and deeply several times, and being sure to force out all the air.