The whole idea behind plugins is that they are added functionality that is theme-independent. The plugins themselves shouldn’t have anything to do with the theme in use. In theory.
In practice, you could run into any or all of the following (or potentially other issues I haven’t thought of in doing some quick thinking about this):
(1) Theoretically, the theme developer could have written a plugin/ integrated it into the theme but as a plugin rather than theme code that would totally break. This is highly unlikely but people do weird things so I suppose it’s possible. If your plugins are available in the repository/ separately from the theme, it’s not going to be a problem.
(2) The actual display of whatever feature the plugin enables could be written into the theme. For example, if it displays a particular widget, it’s possible that that widget was added/ displayed in the theme via a shortcode or template tag, in which case it wouldn’t appear in your new theme until you add it there as well.
(3) Similarly to #2, your new theme may not have the same menus/ number of menus/ widgetized areas, so the display will be different in that regard.
Those are the major things that come to mind. Also if you’re using an older version of WordPress you could lose your widget settings, although newer versions preserve those for the most part.
