I am planning on making a service that uses a modified open source software. Would it be legal for me to profit from it? I am not sure what information would be relevant to this so if I have missed some relevant info, please tell me what additional information is required to determine this.
Yes, you can use open source software commercially. The Open Source Initiative (OSI) says:
Can Open Source software be used for commercial purposes?
However, note that commercial is not the same as proprietary. If you receive software under an Open Source license, you can always use that software for commercial purposes, but that doesn't always mean you can place further restrictions on people who receive the software from you. In particular, copyleft-style Open Source licenses require that, in at least some cases, when you distribute the software, you must do so under the same license you received it under.
When using code from a copyleft license, the question then becomes whether you need to share your derivative source code. This depends on which copyleft license was used and what your program is doing with the code, as some licenses have what is known as the SaaS loophole which would allow you to use copyleft code server side without needing to release your code.
See also OSI’s list of open source licenses.
It depends on what license you obtained the open source software under.
Some open source licenses require you to not use the software for commercial purposes. For example, the CC BY-NC 3.0 license requires that you do not use the software (including derivatives) for commercial purposes.
Some open source licenses, on the other hand, explicitly permit the software to be used for commercial purposes, with some requirements. For example, the MIT license explicitly permits the sale of the software.
Some open source licenses, such as the GPL, have requirements that make it very difficult to sell the software directly (for example, because anyone who obtains a copy of it gains the legal right to redistribute it to anyone for no cost), but it is permitted under the license to sell derivative programs, with certain restrictions (for example, the GPL requires you also make the source code available, and if you edited the source code, the GPL requires that you indicate such, and make the edited source code available).
In order to determine whether you are permitted to profit by providing a service using a modified open source software, you should read the license under which you obtained the software. It is also prudent to get the aid of a lawyer so as to make sure you don't misunderstand the license.