While I really admire the answer from @hoge both for a very concise technique and a wrapper function to generalize it and I encourage upvotes for it, I am compelled to comment on the other two answers that use temp files (it gnaws at me like fingernails on a chalkboard!).
Assuming the file is not huge, you can force the pipeline to operate in discrete sections--thereby obviating the need for a temp file--with judicious use of parentheses:
(Get-Content $file | Select-Object -Skip 1) | Set-Content $file
... or in short form:
(gc $file | select -Skip 1) | sc $file
与恶龙缠斗过久,自身亦成为恶龙;凝视深渊过久,深渊将回以凝视…