Burt leaves after five seasons as a joint manager or sole manager, while Cox has stepped down after joining Burt as joint manager in December.
Burt feels it is the right time to take a break from the game and let the new chairman – who will replace the out-going Mathew Major – appoint his own boss.
Mussels sealed their survival in County League Division 1 with a 5-3 win at Hailsham on Thursday, after they had been reprieved from relegation – after finishing in the bottom three – in the previous two seasons owing to ground grading issues.
Burt said: “The last three seasons has taken its toll a bit, struggling at the wrong end of the table, and I just feel it’s the right time to take a break and spend some time with my family.
“I’ll re-charge my batteries, although will obviously miss it enormously as I’ve been playing or managing for the last 27 years.
“There’s been ups and downs in my time here. I came back here to finish my playing career and then managed with Darren Donnelly, Sammy Donnelly and Jason Rutherford, before Lee came in this season.
“The second half of this season has probably been the highlight of my time as manager – although we did win the John O’Hara Cup in my first season.
“The togetherness and attitude of the squad the last seven or eight weeks has been brilliant.
“We let in 106 goals this season, which was hard to watch, but going forward we were a match for anyone and I always got my team to play in the right way.
“I’d like to think that the legacy of my time as manager here was that we played good football. We sometimes played a bit too much football but that’s how I wanted my team to play.
“I think that’s the way the game should be taught to kids as it’s the right way to play, but around 70 per cent of teams in the league just launch it forward and feed off the scraps.
“That’s not how I wanted my teams to play, so we always tried to get the ball down and play good football.
“I thank everyone at the club but I think it’s right for the new chairman to come in and appoint his own manager and I wish whoever takes over all the best.”
Cox added: “We’re quietly pleased with how this season went in the end and it’s nice to stay up on merit.
“I shall miss being involved and I enjoyed my time at Shoreham. It was nice to finish on a high note and stay up.
“I’ve been playing football since I was 12 and I’m nearly 50 now and have been involved every single season.
“I’m just going to explore other avenues and other sports and I’m quite into my snowboarding, so I’ll go and do a bit more of that.
“Mark and Mathew will both be missed as they were the heart and soul of the club but now is the time for someone else to come in and try to push the club on.”
Burt and Cox both came on as second-half substitutes in Mussels’ 1-0 defeat at Crowborough in their final match of the season on Saturday. Match report on page 106.