What's wrong with my berries?

I live in zone 7 and have some berries planted in 20gal containers. The varieties include red, gold, and purple raspberries 1 blackberry and 2 tayberries. Currently my blackberry and tayberry have developed an issue that I can’t figure out. And my gold berriy that was growing very well is starting to show similar signs. I have all my berries planted in a mixture of peat moss with compost and pine needles as mulch. I pre fertilized the soil prior to planting. I can't plant these in ground because it's a rental property. I water when the top soil is dry so about every 3 days now with the heat of summer. I am desperate to not lose these berries.

Hi Carina and welcome to permies!

What sort of fertiliser did you use? The browning and dieback at the leaf margins is a telltale of salt concentration. Berries usually don't tolerate salts and if the fertiliser is too concentrated, this is a common symptom.

I would try watering them heavily to really flush out the root zone. Use rainwater if possible, especially if you have hard tap water where you live. Let the soil drain well but not dry out between waterings, as the drying can concentrate soluble salts.

I second Phil's idea.  Sometimes too much potassium from the fertilizer will cause that.  

I'd catch the water you're running through the pots in a bucket and use it on other plants.  It's still perfectly good water with good things in it.

Sometimes there are air pockets that get into potted plants, and if the water runs out really quickly that means the soil in the pot is not getting saturated enough.  That may not be your problem right now, but eventually it may cause drying issues around the roots of any plants in pots.  Sometimes after the first watering in of a potted plant the soil shifts and air pockets form.  Use a gloved finger to insert into the soil in various places, and if it finds anything that feels like an air pocket, gently tamp it down, particularly around the sides/edges.  

I put saucers under all pots so the water stays there and the soil in the pot can soak it back up.  I've got the saucers up on little pottery legs that keep it all up off of whatever the surface is.  Saucers also keep deck wood from rotting or water staining concrete, bricks, etc.  Although I just tried to get a large terracotta saucer, which used to be pretty inexpensive, and they wanted $49!!  When did that happen???

That sounds like a wonderful planting medium for blueberries.  They like extremely acid soils that other plants can sometimes struggle in.  Our soils here are have a PH of 7.8 and wild blackberries thrive.  Looking online it looks like raspberries aim for around 6.  In pots it might actually be worth the effort to add some lime if you test your PH and find the number really low.

This is not to discourage flushing the soil in case of salt build up, but it's another thing you can look at.

One of the reasons we put so much emphasis on organic matter in the soil is it helps keep the soil ph in a range where most plants thrive.  Unfortunately, while organic peat is the dried remains of a bog plant.  One of the reasons so many archeological discoveries are found in bogs is they are so acidic the slow the natural decay processes.

For more specifics, I added a lot of peat moss, peat and sheep compost, and composted mulch soil conditioner I purchased from Lowes. To all of this I added lime and berry fertilizer. Maybe I did make a mistake and shouldn't off added more fertilizer? I live in ABQ so no rain for use until the monsoon season this summer. But I just installed an osmosis water system in the kitchen. Could I maybe use this filtered water to flush out the pots?

I suspect the combo of sheep manure and fert might have been a little much. RO water from your house system is great. Cristo's suggestion of capturing the runoff from the drenching is a good one, as is checking for dry pockets in the root zone.

Interesting thing with blueberries is that they tolerate acid soils more than other plants. They are quite happy in normal soil too.
Dan Kittredge talks about this. The plant will create its own optimal soil conditions provided that the soil life is healthy and abundant.