If the Himalayas are any indication, taller mountains in a tropical or near-tropical latitude mean titanic rivers
Yes, it does! Large mountains cause orographic rainfall (as the previous answers tried to explain). HOWEVER, the rainfall is only on the side that shares the source of moisture. In the Himalaya's that means India and Nepal receives a heck of lot more rain than Tibet. It is also one of the reasons why one of the major rivers of the world flows through India. The higher the mountain range the less chance of moisture making it over the barrier.
Take another less specific diagram...
Now, Take the above image and flip it around (I couldn't find any good images from the right perspective). This will be your mountain range you have extended right across the length of the African East Coast. What you have done is effectively cut off any source of moisture from the warm Indian Ocean. You even created a large mountain range on the southern west coast, so any little moisture that might be brought inland by the Atlantic fogs hitting the Namib area won't reach the interior either. So the southern Africa Interior region is going to be DRY.
I got an image of the average long term rainfall of Africa to show you. (Hey, look, 2D!)
You can see that Africa has a large amount of rainfall in the equatorial region due to the Hadley Cell convection system that you correctly referenced in one of your comments. But you can also see that, that southern and eastern Africa are reliant on the Indian Ocean for moisture. If you plonk down a huge mountain range with no breaks, all that moisture is going to flow right back into the Indian Ocean and never make it into the Interior. Rek's suggestion of breaks in the mountain ranges will help channel the moisture into the Interior. Think of it sort of like a vacuum effect, with the dry interior sucking in the moisture laden ocean and mountain air (I actually have no idea if that would work like that, but it sounds cool). I do know that, the moisture will make it into the interior if you have portion of the mountain range have a lower height in some regions.
Now that you have taken the Indian ocean out of the running for a moisture source, you are entirely reliant on the West Africa currents to bring in new moisture.
You will be reliant on the cold Benguela current flowing up the West coast of Africa and warming up. That is a lot of pressure for a single ocean current! Cold ocean currents aren't the best for creating moisture laden air currents. Why? Because generally air flows from warm high pressures, to low cooler pressures; most of the winds will be blowing offshore on the South-West coast. You can actually see that in the map above, where the Benguela current is pushed off to the west, allowing the smaller Angola Counter current to flow south. Oh, and don't forget you blocked off the west coast with your ladle-shaped mountain range... You also extended the Atlas mountains in Morocco and the North West Coast of Africa. You are effectively cutting off any moisture from the ocean currents in that region.
So. ALL of your moisture is equatorial bound. From both the Hadley cell convection and the equatorial counter current. You would think with such a large equatorial ocean current you would be sorted. However that current is in the doldrums. An area with little to no air flow. Sailors used to get stuck in the doldrums for days and weeks with very little wind to fill their sails. Not a lot of that moisture is going to make it's way into Africa. Instead, it flows westward and feeds the Caribbean Current which feeds the Florida Current which feeds the Gulf Stream which brings nice 'warm' and 'moist' weather to the Northern Hemisphere. This current then gets fed by the south flowing frigid arctic currents, cools down and makes it's way back down the north west coast of Africa. What am I driving at? Most of the moisture from the current flows away from Africa.
But I can see you coming back at me with your Lake Congo and Lake Chad. I'm not too familiar with their origins, so I had a quick peek at Wikipedia. The Congo RIVER, the second largest river in the world, is feed by the rivers from the EAST coast of Africa by the East African Rift. But, hold on, didn't you make those Eastern mountains higher? so, less water will be flowing down the western flanks of the mountains? so, um, I hate to say it, less water is available for your Congo River...and hence...no Huge Lake Congo. Sorry. The accepted source of the Congo River is Chambeshi River in Zambia...Rek's solution of a mountain break in the Zambia region would help provide a source of moisture for the Chambeshi.
Mountain Breaks are completely natural. A mountain range is not a constant elevation throughout it's length. Nature has a way of eroding along the lines of least resistance and natural mountain passes form. The Himalaya's are huge, obviously, but have you ever stopped to think how birds migrate every year across them? They can't fly over the very top of them, as the air is too thin for most of them. Bird's use the passes to fly 'over' the mountains.
One point to consider in your scenario is the elevated mountain range itself. This will lead to snow covered mountains in a dry environment? Most of the moisture will be wrung out of the atmosphere at lower altitudes on the Eastern flanks of the mountains, allowing only a small portion of it to form as snow on top. This could provide some spring melt as a source for your interior rivers.
That's it for now. I'd be very interested in the climate of the Eastern side of your African Mountain range! very lush, I'd imagine! Cloud factories, monsoons, rainforest vegetation, probably a few bird's of paradise...
EDIT TO INCLUDE INFORMATION FROM YOUR COMMENT ABOUT THE TETHYS SEA ANDRELATED QUESTION (comment, you really should link your questions together)
Also including extra information on southern African weather for the New Lake Makgadigadi
Ok, I do think you should have mentioned this in the original Post but you still have time to edit. I'll probably just do a brief summary here and answer more fully in your Sahara Savannah question as it's more relevant to that area.
The fact that there is a sea and not land in the region of the between the Arabian peninsular and asia means that super moist air from the monsoon should be able to make it's way into the interior of Africa through Egypt, Sudan and Ethiopia.
I had a lot of time on my hand tonight, so I created a ROUGH GUIDE to John's ALTERNATE EARTH ocean & wind current map. I put all the maps you have provided together. It was on paint, so it is a bit (alot!) rough but should get the rough idea across. This is assuming Madagascar is in the normal place (but I think I saw the very tip of it near Saudi Arabia, a lot more north than current world. you can alter accordingly). This is also assuming that everything else is the same (but knowing you, it probably isn't). I made a educated guess how your altered Himalaya's were orientated. The ITCZ is just a best guess in this new world. Feel to free to amend if you think I got the converging winds wrong.
So what do these maps prove. Well for one thing, it shows everything is connected, which is why I was asking for a larger map to answer your focssued questions. You should get monsoon rainfall in Saudi, Egypt , Sudan and West Ethiopia (actually thinking about it, you get monsoon weather on either side of the mountain range in Ethiopia). So you have a new source of water for your Lake Chad and Congo. Your Red and Dead sea are almost certainly not Red or Dead anymore. They will likely form another large lake/inland sea.
But focussing on Southern Africa.
If Lake Chad and Lake Congo are depressions with the southern end of Africa raised as shown in Rek's 3D map, then lake and river water will not flow south. Rather the water will flow back into Lake C&C. But if the Southern end of Africa is lowered slightly (not the mountains themselves, just the base level of the interior) then you can have lake and river water flow from Lake C&C into Lake Makgadigadi. That is some of the reason why Lake Makigadigadi withered away, as the southern end of Africa was raised by tectonic activity.
Something important to note, I'm not 100% certain, but I have a feeling you will have created a scenario where you have an almost constant Kalahari High Pressure system in the region, throughout the year. Evaporation is going to be high. Like I said before, the air is going to be very dry.
But again, I have thought it through and I think your mountains can save you from the problem of, um, your mountains. With the Elevation being higher than this world, your winter low pressure systems from the Antarctic are going to create some hectic snowstorms in the Cape regions, due to orographic lift. The Cape Town region will no longer experience a Mediterranean climate. Rather I think it will be similar to New York and Boston. Hot in summer, with not too much rainfall, but more than it currently experiences. It will have very cold and snowy winters. This massive amount of snow will then melt and run off into the interior. So while it is dry, you will have constant surface water sources from the south during Spring and Summer (from the snowmelt) and from the north during winter (monsoon supplied Lake C&C).
That wasn't a typo up above with the winter lake supply. Water takes a few months to flow all the way from one end to the other. The Okavango delta (this world's remnant of Lake Makigadigadi) is supplied by rain that falls in the Angola Highlands a MONTH beforehand. You asking water to flow all the way from Egypt and the Sudan through Lake C&C. Expect a time delay.
So in southern Africa, you won't have an even spread of greenness. I believe anywhere not near a water source is going to be dry, dull and almost lifeless. You will have your oasis, river fertilised areas, and lake habitats. sound familiar? I think you have effectively switched the North East African Desert to the South. I don't think you will have vast areas of 'Serengeti' but maybe in isolated valley/basin pockets.
Hope this edit helped.
Note: I was about to add a comment about the mountain range being a volcanic chain, which would naturally have a series of mountain passes - allowing moist air to flow into the interior. But I see you actually said a 'SUBDUCTIVE volcanic mountain chain!'. As in the Andes? and the Marina trench? I did not take into account the deep ocean on the East coast of Africa, I was working with Shallow, Warm water. That will totally mess with the water temperatures, therefore the atmospheric pressure and therefore the winds. There is no real-life example of subductive mountains on the East coast of a continent.
It technically negates my, and everyone's, entire answer!
Which makes me feel really bummed...as I thought I had it mostly solved for you! Next you are going to say, the wind blows in the opposite direction :)