Google-sheets – Conditional formatting for passed dates

With New Google Sheets please try Custom formula is:

1. =today()-a1>7 with Background red and Range: to suit
2. =and(A1-today()>0,A1-today()<28) with Background green and same Range:
3. =A1-today()>=28 with Background white and same Range:

You're on the right track, just get rid of If: the custom formula should be

=And((E2+7)>Today(), H2="No")

Explanation

If requires at least two arguments: condition and the result to be returned. Here you just have a condition, so if is neither suitable nor needed. The formula used for conditional formatting should return a boolean value (True/False), or a numeric value, which will be coerced to boolean as follows: 0 will be interpreted as False, nonzero as True. The formatting is applied if the formula returns True.

In your case, and returns a boolean value, which is all you need.