The fans group hope to complete a takeover of the club that will save it from liquidation within weeks but chairman Murray is concerned Brown's departure could take away vital focus from the all-important rescue bid.
It was announced on Tuesday night that Gary Locke's assistant would not have his contract renewed at the end of January, while managing director David Southern and director of football John Murray have both put their hands up to accept wage cuts.
But former boss Jim Jefferies has criticised the decision to cut his old friend Brown loose and raised the question of whether cash-strapped Hearts even need a director of football in their current predicament.
Now FoH chief Murray has urged fans to forget about the current row - as well as the debate about relaxing the Gorgie's side transfer ban - and concentrate on saving their club.
Murray - who also announced that FoH have appointed Dundas & Wilson as legal advisers and accountants Scott-Moncrieff as its nominated corporate financial advisor - said: "The start of a new year brings optimism to Heart of Midlothian, but at the same time there is a reminder for all of us in events this week that there is much to be done before supporters can see our club exit administration and begin to build for the future.
"It is critically important that all eyes remain on the prize, securing Hearts' survival, and so I strongly encourage everyone involved in this process to remain highly focused.
"Some of the discussion of the last fortnight about the manager, his assistant, and player registrations are understandable, but we must remember why we are all here.
"I am extremely heartened to report that January contributions via direct debit have held up as strongly as every month before. We continue to see over 7,600 contributors and a 99% collection success rate.
"We are all acutely aware of the need to see Hearts exit administration, and plan for the future. We know there is frustration from supporters, but I can assure them we are working every hour to expedite the process. We are ready to move extremely quickly, once matters in Lithuania can be resolved. Hopefully that moment will be soon."