The charity is mentioned in a warning issued by the commission to charities yesterday about the risks of getting involved in arrangements with private landlords and companies that take advantage of business rates relief for charities.
The warning also mentions the case of the Public Safety Charitable Trust, which, after a High Court ruling last week, could be liable for almost £2m in unpaid business rates to three local authorities, and is subject to a Charity Commission regulatory compliance case.
The regulator also opened a statutory inquiry in April 2012 into the Kenya Aid Programme to examine regulatory concerns about potentially significant financial loss to the charity and "serious governance failures".
The charity is in dispute with Sheffield City Council, which won a liability order against it in Sheffield magistrates court in October 2011. After an appeal by the charity the High Court ruled in January this year that the decision, which held that the charity was liable for a total of £1.6m in unpaid business rates, was "flawed" and the district judge should look at it again.
The case will be heard at the magistrates court on 24 June.
The charity rents two industrial units in Sheffield for storing furniture that is sent to schools in Kenya.
The charity leases the properties for a peppercorn rent and its landlords pay an annual donation to KAP in return. It had an income of £80,000 in 2011.
Like all charities, it is entitled to an 80 per cent discount on council business rates paid if the building is used "wholly or mainly for charitable purposes".
But the council claimed that KAP was not eligible for the relief because it did not use the two properties wholly or mainly for charitable purposes and said it owed the total of more than £1.6m on the two properties between September 2010 and March 2012.
Maurice Smith, chief executive of KAP, said: "We are a small charity but Sheffield Council has refused to give us business rate relief. Instead of fundraising, we get donations from the landlord and use the money to help people in Kenya."
Sheffield City Council declined to comment on the case while it was ongoing.
The commission said in a statement that such arrangements could "represent a significant risk for charities and trustees". Trustees could find themselves personally liable for any debts, the regulator warned.
Michelle Russell, head of investigations and enforcement at the Charity Commission, said the regulator would take "firm regulatory action" in cases where charity trustees were not exercising their duty of care and taking "proper decisions".
"The commission has received information from a number of local authorities concerned about situations where charities are entering into tenancy agreements on commercial properties but where in practice the properties are, or appear to be, empty and/or only minimally used," she said.
She said the regulator was concerned that these charities could "find themselves involved in what local authorities might consider to be business rates avoidance by landlords".
"As seen in this case involving the Public Safety Charitable Trust, this could result in charities losing not just the discretionary discount, but being required to pay full business rates."