There is a problematic assumption in this question: you are moving the continents around on a mercator projection, which is very different from adjusting the earth by tilting it 'southwards' (basically, rotating the earth sideways in space).
This might put the biggest landmass (Asia) further south, but would also move the continent on the other side of the globe (North/South America) further north.
Let's say you're some godlike entity. You stop time at noon in Islamabad (roughly east-west center of the eurasian continent) on the 21st of June. Then, you 'move' the earth's equator so that the new equator touches the two tropics (the northern one 10 degrees below Islamabad, the southern one roughly 3000km in the ocean west of Chile). You change the earth's spin so that it now spins around its new axis, shifted 23 degrees north, changing the astronimical date from 21st of June to 21st of October (or March).
The half-way point of these coordinates would be the equator on the Atlantic ocean, in front of Liberia (17 degrees west, 0 degrees north). Everything up to 180 degrees west of this point would move northwards, everything east of this point moves southwards.
On the balance of things, it would create more agricultural land. In a human timescale though, such an alteration would have devastating consequences for the life currently on earth, which can't just 'pack up and go' somewhere else. If it happened over a few thousand years, no problem. New virgin rainforest and arable soils require hundreds of years to form. The Gobi and Sahara won't turn into lush rainforest even if it started raining a lot there for quite some time.
In contrast, a bunch of existing rainforest (the Amazon?) would move into the horse latitudes, become starved off its water supply, and mostly die off in a manner of years.
This is the problem with climate change: not necessarily that the climate changes, but how fast it happens.
If you're a bit more creative than that, there's ways you can tilt the earth for more forest climates, here's an example of what someone managed to do:
https://www.reddit.com/r/MapPorn/comments/3zmrc1/the_biomes_of_a_tilted_earth_oc_1204x840/