How to remove the first character if it is a specific character in SQL

Here is the code and a SQLFiddle

SELECT CASE 
WHEN substring(telephone_number, 1, 1) <> '9'
  THEN telephone_number
ELSE substring(telephone_number, 2, LEN(telephone_number))
END
FROM Telephone

While all other answer are probably also working, I'd suggest to try and use STUFF function to easily replace a part of the string.

UPDATE Telephone
SET number = STUFF(number,1,1,'')
WHERE number LIKE '9%'

SQLFiddle DEMO

I recently solved a similar problem with a combination of RIGHT(), LEN() & PATINDEX(). PATINDEX will return the integer 1 when it finds a 9 as the first character and 0 otherwise. This method allows all records to be returned at once without a CASE WHEN statement.

SELECT
    RIGHT(number, LEN(number) - PATINDEX('9%', number))
FROM Telephone

Update Telephone set number = RIGHT(number,LEN(number)-1) WHERE number LIKE '9%';