Plugin and theme conflicts are unfortunately a fact of life with WordPress. Contact Form 7 plugs correctly into the WordPress architecture so that it does not cause conflicts with other plugins and themes.
However there are many plugins and themes that are not so well written. Like many other popular plugins, CF7 is frequently reported as not working in the WordPress Forums when the real cause is conflict from another theme or plugin.
Step by step approach to finding Plugin & Theme Conflicts
1.) Disable (temporarily) all plugins other than CF7
a.) Does it work now? – Turn back on plugins one by one until you find the conflict
b.) You may need to clear the browser cache between each one
2.) Disable (temporarily) your current theme and load one of the WordPress Default themes
a.) Does it work now? – The problem is with your current selected theme
3.) With everything disabled, try (if you can) to correct the problem by reinstalling elements that may have become corrupted.
You may need to :
1.) Deactivate Contact Form 7 and completely delete it. Then re-install it into your WordPress instance.
2.) Reinstall your theme.
3.) Delete and reinstall the plugin causing the conflict.
4.) Re-upload your wp-content folder with a fresh copy.