If it is necessary to obtain permission to use quotes from other sources, it is the author’s responsibility to do so. When the quote is to be used in a magazine, seek the permissions before you submit the article to a publisher. Any permissions needed for a book should not be requested until you have a buyer for the book, since permissions are often given to a particular publisher for a specific project and may expire after a certain time.
If you are not sure if certain quotes require permission, your publisher can usually advise you. Book publishers often have a rights and permissions department that reviews each manuscript for that purpose.
Be aware that some authors and publishers grant permissions for quotes only if you pay a fee set by them. If such a fee is charged, you will be responsible to pay it—not the publisher—so you will have to decide if the quote is worth the asking price. If not, delete or replace it.
With many publishers, the need for permission, or the fee set, are based on how important the quote is to your article or book, and how predominant a place it will have in your work. In other words, if it is just a quote within the text somewhere, it is more likely to be considered fair use. But if it is going to be highlighted, printed in a box or used to introduce a new chapter, they may want payment. The grim reality is that at many houses, you will be asked to pay for the quote—even if it should be fair use—just because you asked. For that reason, it does not pay to be too cautious and ask permission for every quote—whether you think you need to or not. You could end up paying much more than you need to. It is also encouraging to note that few infringement cases are brought against authors of books and articles—most have to do with TV, drama, movies, etc. where much more money is involved. At the same time, don’t let the slim possibility that you will be sued deter you from making wise and informed decisions about when you should or should not ask permission. Common sense may be your best guide.