What can I substitute "shall" with in these sentences to sound less formal?
You look cold. Shall I put the heater on?
What shall I bring you, tea or coffee?
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You could say to a friend:
You look cold. Do you fancy the heating on? (very informal)
You look cold. Do you want the heating on? (informal)
You look cold. Would you like the heating on? (less formal)
and:
Fancy a cup of tea? (very informal)
What / How about a nice cup of tea or coffee? (informal)
Would you like tea or coffee? (less formal)
"Should" is likely the best candidate for your purposes. When used in this context as an auxiliary verb it is less formal than "shall", but the meaning is nearly the same and the sentence would need little to no other changes.
You look cold. Should I put the heater on?
What should I bring you, tea or coffee?
This worksheet should help to learn more about the similarities and differences between "shall" and "should".
Alternatively, you could use a phrase to replace "shall" and restructure the sentence a bit to retain more of the original meaning. For example:
You look cold. Would you like me to put the heater on?
"Shall" is very uncommon in American English, and is considered very formal when it is used. I believe it is much more common and has less implication of formality in Britain. So first consider your audience, it may be that "shall" would be just fine if you're in the UK.
In America, you could use any of these in informal situations:
Should I put the heater on?
Do you want [me to put] the heater on?
Would you like [me to put] the heater on?
Can I put the heater on for you?