What would be the effect on the economy if all VAT / sales tax was removed, but income tax was increased to compensate so that the government has no change in income?
In the long run there would be no difference as income will always eventually equal consumption so in the long run, flat 30% VAT tax would be equal to flat 30% Income tax. This is so because income can be only consumed or saved but saving is just deferred consumption. Assuming away some dynamic changes in taxation in future it does not make any difference from pure economics perspective.
However, in practice it is much better to have multiple sources of taxation instead of one. Having two sources of taxation helps combat tax avoidance (some people might easily avoid income tax like for example baby sitters or tutors but they can’t just as easily evade VAT). Also, in cases you care about redistribution it’s worth noting that it is very difficult to redistribute income with VAT if you apply it to individual products and not to total spending (which is virtually impossible in real life). So in cases where you want to have progressive non-linear tax to redistribute income, income tax is far superior instrument to VAT.
There is also some recent empirical evidence that shows that they might not be equivalent and which suggest labor taxes have higher welfare loss (you can see one example of such evidence here)). However, I would take the paper with a grain of salt. As the paper itself states it’s considered well known result in public economics that the two taxes are equivalent and laboratory experiments can have they own problems, and it’s hard to say what would be a convincing reason why they should not be equivalent. Still it’s worth knowing that the result is being challenged.