New housing starts, in a third consecutive monthly increase, rose 12 percent in August to an annual rate of 1.4 million units, the Commerce Department reported yesterday.

The rate was up 54 percent from the recession low of 906,000 units in May, but some economists warned the improvement in the housing market likely will be halted by rising interest rates.

Separately, the Federal Reserve Board reported that the nation's factories operated at a seasonally adjusted 74.5 percent of capacity in August, up slightly from the 74.4 percent utilization rate in July.

Mortgage interest rates now are between around 13 1/2 percent and 14 percent in most parts of the nation, up more than 1 percentage point since the middle of last month. The higher rates "are simply going to kill housing," Michael Sumichrast, chief economist for the National Association of Homebuilders, warned.

Sumichrast said the rise in rates probably means that August was the last month for some time which starts will be higher than the month before. However, other August figures from Commerce showed a continued increase of nearly 100,000 units at an annual rate in the number of building permits issued, usually an indication of the future course of actual starts.

The recovery in homebuilding activity from the May low was sharper than most economists had expected, largely because mortgage interest rates fell much more rapidly than normal after the recession took hold. The fast drop was due primarily to the new way in which the Federal Reserve intervenes in financial markets to achieve its targest for growth of the money supply. The procedures put in place last October allow much greater swings in interest rates, which produce a rapid adjustment downward one the recession hit.

Now the shoe is on the other foot, at least temporarily, with rates rising much sooner than normally would be the case. In fact, some economists are not sure the economy might not turn down again in coming months.

Most of the gain from July's revised rate for starts of 1.2 million units came in the single-family category. Starts of that type of house rose from an annual rate of 870,000 units in July to 974,000 in August. Starts of structures with two, three, or four units climbed from a rate of 81,000 in July to 134,000 last month. The number started as apartment houses with five units or more fell slightly from a rate of 298,000 in July to 291,000 in August.

Regionally, the largest increase in starts came in the West, where units were begun at a 360,000-unit rate in August, compared with 278,000 the month before.

Even with the gains of the last three months, the total number of starts remained 22 percent below the rate for August 1979. During the first eight months of this year, 787,800 housing units were started compared with 1.2 million units in the same period in 1979, a 34 percent decrease, the department said.

The federal Reserve said the increase in factory operating rates was the first since September 1979. A year ago, the rate for primary processing industries was 17 percentage points higher, while the rate for advanced processing industries was 7 percentage points higher.