Scripture gives no instruction on this matter. One will search the Word in vain for a command from the Lord regarding how the Bible is to be read. Scripture almost always focuses on the issue of the heart, rather than the form one uses. In other words, sitting or standing isn't nearly as important as the receptivity of one's heart to what is being read.
With that said, the Bible does offer us some evidence. While it doesn't proscribe a method for reading scripture, it does describe what some of God's people did in Bible times when reading the Word. Nehemiah chapters 8 and 9 give us our most direct evidence. Nehemiah 8:1-5a states:
"All the people assembled as one man in the square before the Water Gate. They told Ezra the scribe to bring out the Book of the Law of Moses, which the LORD had commanded for Israel. So on the first day of the seventh month Ezra the priest brought the Law before the assembly, which was made up of men and women and all who were able to understand. He read it aloud from daybreak till noon as he faced the square before the Water Gate in the presence of the men, women and others who could understand. And all the people listened attentively to the Book of the Law. Ezra the scribe stood on a high wooden platform built for the occasion. Beside him on his right stood Mattithiah, Shema, Anaiah, Uriah, Hilkiah and Maaseiah; and on his left were Pedaiah, Mishael, Malkijah, Hashum, Hashbaddanah, Zechariah and Meshullam. Ezra opened the book. All the people could see him because he was standing above them; and as he opened it, the people all stood up." (NIV)
In this passage we notice first and foremost the condition of the people's heart. "They listened attentively" as Scripture was being read. We also see something of the form they used. Both the minister (in this case, Ezra) and the other religious leaders stood on a high platform when reading the Word. The people, seemingly in unison, stood when Ezra opened the scroll. This unprompted standing is best explained if we see this as the accepted custom when God's law was being read. Nehemiah 9:3 makes mention that they would stand for long periods of time, even up to 1/4 of the day. Obviously, they took the reading of the Word seriously. Of course, this was the assembly of the nation, so one could hardly expect them to have the ability to provide seating to such a huge crowd of people.
On the other hand, we also see the reader and crowd sitting during Scripture reading in Jeremiah 36:8-14. In that passage Baruch is summoned to read to group of political officials. Verses 14-15 state that they told Baruch to "bring the scroll from which you have read to the people and come." So Baruch son of Neriah went to them with the scroll in his hand. 15 They said to him, "Sit down, please, and read it to us."
In the end does it matter? As with most things in life, God judges the heart (by the way, that doesn't lower the standard, it raises the standard). Each congregation should determine the practice that best promotes a spirit of worship. Often this would include standing (I do wonder why we are so willing to stand for a contemporary worship song but get annoyed when asked to stand for Scripture reading...could that be revealing a heart issue?). Perhaps it would include sitting comfortably to help aid concentration. Either way, God doesn't really speak to this issue.