The IFRAME element may be a security risk if your site is embedded inside an IFRAME on hostile site. Google “clickjacking” for more details. Note that it does not matter if you use <iframe> or not. The only real protection from this attack is to add HTTP header X-Frame-Options: DENY and hope that the browser knows its job.
If anybody claims that using an <iframe> element on your site is dangerous and causes a security risk, they do not understand what <iframe> element does, or they are speaking about possibility of <iframe> related vulnerabilities in browsers. Security of <iframe src="..."> tag is equal to <img src="..." or <a href="..."> as long there are no vulnerabilities in the browser. And if there’s a suitable vulnerability, it might be possible to trigger it even without using <iframe>, <img> or <a> element, so it’s not worth considering for this issue.
In addition, IFRAME element may be a security risk if any page on your site contains an XSS vulnerability which can be exploited. In that case the attacker can expand the XSS attack to any page within the same domain that can be persuaded to load within an <iframe> on the page with XSS vulnerability. This is because vulnerable content from the same origin (same domain) inside <iframe> is allowed to access the parent content DOM (practically execute JavaScript in the “host” document). The only real protection methods from this attack is to add HTTP header X-Frame-Options: DENY and/or always correctly encode all user submitted data (that is, never have an XSS vulnerability on your site – easier said than done).
However, be warned that content from <iframe> can initiate top level navigation by default. That is, content within the <iframe> is allowed to automatically open a link over current page location (the new location will be visible in the address bar). The only way to avoid that is to add sandbox attribute without value allow-top-navigation. For example, <iframe sandbox="allow-forms allow-scripts" ...>. Unfortunately, sandbox also disables all plugins, always. For example, historically Youtube couldn’t be sandboxed because Flash player was still required to view all Youtube content. No browser supports using plugins and disallowing top level navigation at the same time. However, unless you have some very special reasons, you cannot trust any plugins to work at all for majority of your users in 2021, so you can just use sandbox always and guard your site against forced redirects from user generated content, too. Note that this will break poorly implemented content that tries to modify document.top.location. The content in sandboxed <iframe> can still open links in new tabs so well implemented content will work just fine. Also notice that if you use <iframe sandbox="... allow-scripts allow-same-origin ..." src="blog:..."> any XSS attack within the blob: content can be extended to host document because blob: URLs always inherit the origin of their parent document. You cannot wrap unfiltered user content in blob: and render it as an <iframe> any more than you can put that content directly on your own page.
Example attack goes like this: assume that users can insert user generated content with an iframe; an <iframe> without an attribute sandbox can be used to run JS code saying document.top.location.href = ... and force a redirect to another page. If that redirect goes to a well executed phishing site and your users do not pay attention to address bar, the attacker has a good change to get your users to leak their credentials. They cannot fake the address bar but they can force the redirect and control all content that users can see after that. Leaving allow-top-navigation out of sandbox attribute value avoids this problem. However, due historical reasons, <iframe> elements do not have this limitation by default, so you’ll be more vulnerable to phishing if your users can add <iframe> element without attribute sandbox.
Note that X-Frame-Options: DENY also protects from rendering performance side-channel attack that can read content cross-origin (also known as “Pixel perfect Timing Attacks“).
That’s the technical side of the issue. In addition, there’s the issue of user interface. If you teach your users to trust that URL bar is supposed to not change when they click links (e.g. your site uses a big iframe with all the actual content), then the users will not notice anything in the future either in case of actual security vulnerability. For example, you could have an XSS vulnerability within your site that allows the attacker to load content from hostile source within your iframe. Nobody could tell the difference because the URL bar still looks identical to previous behavior (never changes) and the content “looks” valid even though it’s from hostile domain requesting user credentials.