Is it necessary to include fully depreciated improvements on form 4797 for sale of rental property?

Unless you are selling the improvements indivudally, they will be reflected in the selling price of the rental property along with the accumulated depreciation entering into your calculation.

It depends on what you mean by improvements.  If you are referring to an improvement to the real estate itself, you are right. 

But if you are referring to non-real estate items, such as appliances or carpet, it wouldn't really be a wash.

The sale price should be allocated based on the Fair Market Value of each item.  In most cases allocating the sale price would reduce the long-term capital gain on the house (taxed at lower rates) and result in a gain for the non-real-estate items (recapture at ordinary tax rates).

With that being said many tax preparers just ignore the non-real estate items and lump it all together with the real estate.  It isn't correct to do it that way, but it is common.

Or you can do it Tax guy Bill's way which is more correct.

I report the real property, and the appliances etc, as two separate sales.  In fact, Lacerte 'yells' at ya' if you don't if you are using the bulk sale feature.

I usually report as the sales price, the amount necessary to zero out the gain/loss on the appliances etc.  UNLESS there's something significant involved; or if the sales contract details these non-real property items separately with amounts.  I've seen that happen a few (very few..) times in my career.

Of course, you have to reduce the sales price of the real property by the amount allocated to the appliances, etc.

AND if there's a § 1031 exchange involved there's more to think about as the non-real property items can't be part of the exchange.

If you don't include them, the Unrecaptured 1250 Gain component will be understated.