It's important to remember that there are all kinds of taxes, federal, state and local. Since you don't specify, the possible ramifications of removing a tax can vary greatly depending on what the tax is.
So for some different examples than the ones listed above, consider the removal of a local sales tax for criminal justice. While the sales tax is lowered, and this is attractive for new businesses and can stimulate sales, a community can also lose law enforcement positions, crime prevention programs or even available jail space. With a rise in crime, you can negate the positive effects I listed above. So whenever a type of tax is removed, there will be both positive and negative effects. There has to be.
I encourage you to read a little bit about the "Fair Tax" plan - endorsed by Neal Boortz.
This is an idea that the government should get rid of income tax (and the deductions, and the brackets, and the credits, and the once a year returns, and the write-offs, blah, blah, blah) - and replace it with a 23% sales tax on everything. There are a lot of small details, but the basic idea is that by converting the majority of tax revenue to sales tax - everyone is taxed as a consumer rather than a producer. We are taxed on what we deem most important - the things we buy.
Anyway, it is an interesting refute to the idea that getting rid of income tax would cause a national melt-down.
If we are looking at the US federal government, then by far the largest tax is the income tax. If this tax were removed, there would be major implications, even if it were to be replaced by some other kind of tax (as it would surely have to be).
If the income tax were removed, the tax system would surely become more regressive. A regressive tax system is one in which poorer people pay a higher percentage of their income in taxes than richer people do. The income tax is progressive (where richer people pay a higher percentage). If it were replaced with a sales tax, for example, the system would become more regressive.
Having a regressive system would surely make many people angry. They would not like the idea that middle class people were paying a higher percentage of their money in taxes than rich people were. This might lead to even more anger against the government than we currently have.