Can the selections from a multiple choice Text field be narrowed based on logged in user's role?

Yes, that is a problem.

Instead of restricting the view, just group the dropdown by the role.  So you'd have all of one at the top, and the other at the bottom.

Just curious, if a formula text field, is using a value from a restricted lookup, does it still show up, or would it be blocked?  Maybe you can convert the dropdown to a formula text field and display that.

If its a formula text field, there is an option on formula fields to ignore "sub field permissions". Not sure if that solve the problem though.

Actually maybe it does solve the problem.  You could have a formula  be just a mirror of another field and ignore sub field permissions.  this question has been asked before and I have encountered the problem myself and came up with very messy solutions driven by form rules.  It deserves a solution built as a demo app and put into the exchange :)

Thanks for everyone's help on this. I want to try the option with a formula text field, but I'm not sure where to start??

I have re-read your initial question.  Do you have a table of Employees who have Roles?  If you did then you could look up that Role to the Task Table.

Then, you can have a table of Choices for the Task Name with an extra column for the Role.
 
Then, you can set up a conditional drop down for the task name which is conditional on previously having chosen a Role.

So this means that if your current multiple choice field is driven off a basic list of choices in the field itself, instead, then will need to be chosen off a table.

Thanks, I think I have it working now.