Harvesting Sweet Berries with Fortune enchanted tools does not yield an increase in item drops. I have done quite a few tests harvesting ("Use" button) and also mining ("Mine" button) rows of 64 Sweet Berries.

... extensive testing...

(Steps to Reproduce) In a Creative Flat world, commands such as...

/fill ~1 ~ ~ ~1 ~ ~63 dirt
/fill ~1 ~1 ~ ~1 ~1 ~63 sweet_berry_bush ["growth": 3] 

... will produce a row of 64 mature Sweet Berry bushes on dirt blocks in front of the player. 

As far as I am aware, harvesting most crops (including Nether Wart) with Fortune tools will give an increase in drops, acknowledging that Wheat and Beetroot only increase the seed drop rate. I would expect to see the same 50-150% increase when harvesting Sweet Berries.

From my tests, harvesting (Use action, not mining/breaking bush) 64 mature Sweet Berry bushes showed no increase in drop rate with Fortune enchanted tools over unenchanted/empty hands:

At the previous "third stage" of berry bush growth (Age: 2), there is still no increase in drops when harvesting with Fortune enchant:

Mining the mature Sweet Berry bushes (bush is destroyed), revealed different results:

This very small 15% increase in drops while destroying the bushes with Fortune 3, is a very poor return, where I now have to replant the berries to grow more, effectively only returning around 1.8 avg. drops.

Thankyou for your consideration.

Next day edit:

Thinking about it overnight, I conducted more testing today:

For comparison, I tested other crops:
The following crops are harvested by mining the block/plant, which is where the Sweet Berries are different.