Problems with text formats and Iframes
Each text format that you create has some different processing options.

If you select 


  Display any HTML as plain text


Then the result is what you explained, where HTML code is escaped and displayed instead of having any effect.

Instead


  Limit allowed HTML tags


Can be used to limit the kind of html tags which are allowed, but if disallowed tags are used, they will not take effect.

It sounds like you are confusing the two.

Update

The text processing and input filters are not related.

You have filter called plain text, but you could call it anything, like super input field, pony or whatever. When you select filtered input, you get to choose which filter to use.

The other text formatting option is plain text. Selecting this means that you wont use any text filter, not even the one you might call plain text. The result of using this kind of text processor is that the text will be rendered as plain text.