To say that no first edition copies of Sonnets from the Portuguese exist would perhaps be premature; it is entirely possible, even likely, that first edition copies exist in private collections, but their whereabouts are not public knowledge, and there have been no online auctions or sales. Because of this, the most commonly collected copies of Sonnets from the Portuguese, one of Elizabeth Barrett Browning's most acclaimed works, are those printed later that were limited runs or signed by related parties.
Oddly enough, the copy of this book to fetch the highest price at auction recently was a known forgery. Thomas James Wise was a collector in the early 20th century who was exposed in the 1930's for forgery. One of his most famous forgeries was of Sonnets from the Portuguese, with a claimed publish date of 1847, a full three years before it was actually published. Regardless, the scandal was so large that the forgery itself became famous, and it recently sold at auction for nearly $10,000.
Illustrated or beautifully bound printings from the early 1900's are considered collectible, often selling for several hundred to over a thousand dollars. Also considered highly collectible is a printing of the book put out by The Roycroft Shop in 1898 with an introduction by Elbert Hubbard. There are 480 numbered, signed copies of this printing, which depending upon condition can sell for $200 to $500. Several artists have either illustrated or illuminated this book over the years, so copies signed by the artists are always collectible, though most don't sell for more than a few hundred dollars at most.