How do I write awk '{print $1+$2+$3} file1 > file2 for n columns instead of 3?

You want to compute the sum of the fields for each record, so it's just:

awk '{sum = 0; for (i = 1; i <= NF; i++) sum += $i; print sum}' < file1 > file2

The curly braces begin an action statement that is executed on every line of the input; there is no preceding condition that would limit its execution to lines that satisfy such a condition.

On each line:

1. Initialize a sum variable to zero.
2. Loop through the fields, starting at field #1 and ending at the last field (the special variable NF), and increment sum by the value of that field ($i).
3. Print the value of the sum variable.