OK, I have a crazy idea but haven't tried it out, so I have no formula to post in here:
1.) Paste the very first entry (by start date) onto your Team Projects tab. You have to start somewhere. If this is something you cannot do, you can make a false row with false start date that precedes any of the projects. In either case, you have a starter row to build off of.
2.) Next you need to do a competition of the other four tabs using nested IF statements to figure out which has the next earliest start date. So, in row 2 on the projects tab you need to use row 1's starting date, then find the next date after that on Kevin's, Dale's, Dave's, and Brian's tabs and compare each one to the others. ex.: If Kevin's is earlier than Dale's, then if Kevin's earlier than Dave's, then if Kevin's is earlier than Brian's then add that row, else if Dale's is earlier than Dave's, then... etc. That will be a long-ass nested IF statement!
3.) After you set up row 2, though, then all the other rows should just be a copy/paste of that row, except referencing the row above it (row 3 references row 2, etc.)
Will it work? *shrug* It should, I think. You know enough about excel to try it, I suspect.