Can I have a <pre> Tag inside a <p> tag in Tumblr?

You cannot nest block-level elements, such as <pre>, inside <p> in HTML, so Tumblr is behaving correctly. You should close your paragraph before inserting the pre-formatted text.

If you want inline code-style text, try the <code> tag. You can then supplement it with CSS if needed.

<p>I like to use the <code>code</code> tag in my HTML files.</p>

code
{
    padding: 2px 4px;
    color: #000000;
    background-color: #eeeeee;
    border-radius: 3px;
}

Also, I just realized where I was having a similar issue pasting code into tumblr inside <pre> and <code> tags is that just because you wrap your code in those tags doesn't prevent you from having to escape the carets with < and > instead of the carets.

And the HTML5 terminology is:

• p's "Content model" is "Phrasing content": "https://html.spec.whatwg.org/multipage/grouping-content.html#the-p-element
• pre's "Categories" are: "Flow content" and "Palpable content": https://html.spec.whatwg.org/multipage/grouping-content.html#the-pre-element

Since pre's categories don't include "Phrasing content", pre cannot be put inside p.

Maybe it is possible to simulate pre with CSS styles though: How to change a span to look like a pre with CSS? but I'm not sure it is allowed in p.

See also: List of HTML5 elements that can be nested inside P element?