Note: I have not vetted any of the tools below. Remember if you use a service that uploads your file to convert it, your file is stored on a strange server temporarily - so don't use it with anything that is private!
You can always just take a camera/smartphone/tablet photograph of the text you are interested in and attach it to a Zotero record (by dragging it or using the paperclip), but these apps allow you to take a photo of that text with your device which will then be converted to a searchable PDF for you to attach.
Zotero can search inside attached files that are certain file types, notably OCR'd PDFs (note, you must install the PDF indexer in the Zotero preferences for this to work). To me, this is the real killer app for Zotero: it creates the database of you. Converting non-PDF documents to PDF makes them searchable. If you copy an article from a print source it will be an image file that is not searchable. Run it through Adobe Acrobat Professional (or convert it with a similar tool) and now it is a file that can be indexed and searched within Zotero. Mobile apps on your smartphone or tablet make it easy to capture great pages or paragraphs in a book or article--just be sure to also capture the citation! ;-)
What about articles that you receive from Interlibrary Loan? Here at WSU, at least, ILL'd articles can be searchable PDFs or flat PDFs that are flat image files and not searchable. Your options for an image PDF are:
OCR - Optical Character Recognition.
In addition to creating notes about your PDFs in Zotero, you can write directly on them with these tools. You can still print out a clean copy of the original PDF if you need to.
The apps included here can sync with Zotero using ZotFile or the browser version of Zotero on your tablet. It does not include tablet apps such as iAnnotate, PDF Expert, and GoodReader that enable you to annotate PDFs and other documents.
Annotating your PDFs digitally as you read them can be very valuable and an alterantive to printing them out. Some types of annotations might not be searchable by Zotero (and other programs), so you will want to do some testing to see what is searchable.