Linux

via this answer

cat text.html | xclip -t text/html

Mac

via this answer

cat text.html | textutil -stdin -format html -convert rtf -stdout | pbcopy

Windows

In older Windows, you can natively only copy plaintext (via this answer).

type text.html | clip

In PowerShell you can copy rich text:

type text.html | Set-Clipboard -AsHtml

If you create a C:\sandbox\pbcopy.ps1:

type $args[0] | Set-Clipboard -AsHtml

Then you can enable scripts and then run it from anywhere (cmd.exe, .bat files, etc):

powershell C:\sandbox\pbcopy.ps1 text.html

There are a few different Cygwin commands to copy to Windows clipboard and it looks like cygwin provides xclip, so you could probably use the Linux solution on Windows if you have cygwin.