Try http://verseview.info, which I've found works rather nicely. There are a lot of different bible versions around on the web, and if you can program a little and your country's copyright laws allow, you may be able to transform them into the simple XML format that VerseView reads.

At some point I'd love to have a go at creating a VerseVIEW plugin for Screen Monkey. I think it would be hard to beat VerseVIEW's UI for showing a Bible verse live on the fly - just click the book, click the chapter, and click the verse, or then just type "Joh 3 16" and press Enter. On a good day, I can get the verse on the screen before the speaker says its first word :). (And there's a challenge for Screen Monkey: make showing a song on the fly as quick!) But if you know the verses ahead of time from the speaker's notes, it would be great to make a clip for each, and have the whole service controlled from one program. That's an area where Screen Monkey excels, and where VerseView initially had a lot to learn - but all credit to its author for getting a lot closer in v5.0.

VerseVIEW also has a remote control feature, which can be used from a Screen Monkey web clip (thanks for asking your question - I'd been meaning to try this but never got round to it, and it looks like it'll be really useful!):
1) Choose VerseVIEW | Remote VerseVIEW | Enable Remote VerseVIEW, copy the URL shown, then close the dialog
2) Make a web clip in Screen Monkey, using that URL. The web clip shows an HTML form where you can enter a verse reference, present it, show the next verse, and clear the display. The verses are displayed by VerseVIEW using its own graphics and presentation settings, not in a Screen Monkey layer
3) Since we don't want the HTML form to show up on the projector, set the web clip to the Audio Layer(!)
4) You can control the VerseVIEW display from the Live Monitor's Audio Layer tab

A little inspection of the HTML and JavaScript source code of the remote control form shows that we can go a bit further if we want, and make Screen Monkey web clips that immediately have VerseVIEW perform an action: e.g.
5) display a particular verse: /URL/test.txt?command=1&value=Joh 3 16
6) display the following verse: /URL/test.txt?command=2&value=0
7) clear its verse display: /URL/test.txt?command=4&value=0
In all three of these, replace /URL/ with the URL VerseVIEW showed when you enabled its remote control, set the clip to the Audio Layer, and set it to Link | At Start to "Clear Audio Layer" (otherwise the clip remains active, and clicking the next one doesn't work for some reason).

So now we can have a clip in Screen Monkey for each passage we know ahead of time: just make one like in 5) then copy and edit it to change the verse reference. Clips 6) and 7) can be used with all those verses, and we still have clip 2) for displaying a new verse on the fly. Wow! I love it when a pair of programs work together!