They would deal with it. The problem you would have is that formatting almost wholly goes out the window and any signaling with it. Search engines look to the tags for clues as to what the topic is, what the author thinks is important, and how to place the page in the SERPs. Header tags, paragraphs, and the like, are important tools to communicate your ideas just as much as the written word. For example, search engines look to the header tags to know what the topic of the following paragraphs are. Each term used in a tag is weighted accordingly and this helps to match the page to a search intent.
This answer gives you some idea of how this works: Improve Google ranking for general vs. specific keywords There is much more to it than that of course, but it is a good jumping off point.
Otherwise the text would be taken as a single paragraph with breaks in between. This is standard formatting for text to HTML though some parsers would make each paragraph on it's own. It is hard to tell which method applies. There are some plain text pages in Google for example, but it is not recommended. In this case, each succeeding paragraph would have less weight for the terms used and there would be no boost for terms due to the lack of signalling with tags. The page would perform, but not likely well. You see, search is a competitive thing these days. Every piece of content has to compete or it will get lost in the huge mass of data available. Any plain text page, unless highly relevant to a search query, would not be seen. Not any more.