I had the same problem. I had modded shoes that glitched somehow so I couldn't remove them from the dresser to delete them. I solved it, but I had to edit my game save file to do so. Be very careful if you edit your save file! Make a backup!
Your save files are located in the Saves folder in the StardewValley folder that also contains your screenshots. (C:\Users\[yourname]\AppData\Roaming\StardewValley)
Since the 1.5 update, your saves will be in a folder with your farm name. Inside will be a file with your farm name and a bunch of numbers. Mine is "OneFive_268276342" for One Five farm. You need to open this file in Notepad or Notepad++ (not Word). Search for the name of your problem item.
This is the tricky part. You have to delete the entire entry of the problem item. In my case, it was "White Dress Shoes." These are a boots item. The start of the item block looked like this: <heldItems xsi:type="Boots">
You need to select this block of text all the way until you see: </heldItems>
Once you have the entire item entry selected, then delete it. Don't make any other changes.
If you accidentally delete part of the block of another item, you will cause errors. So be careful!
This is the full block that I deleted, for reference (this is from my backup copy).
<heldItems xsi:type="Boots"><isLostItem>false</isLostItem><category>-97</category><hasBeenInInventory>false</hasBeenInInventory><name>White Dress Shoes</name><specialItem>false</specialItem><SpecialVariable>0</SpecialVariable><DisplayName>White Dress Shoes</DisplayName><Name>White Dress Shoes</Name><Stack>1</Stack><defenseBonus>1</defenseBonus><immunityBonus>2</immunityBonus><indexInTileSheet>2032</indexInTileSheet><price>5000</price><indexInColorSheet>103</indexInColorSheet><appliedBootSheetIndex>-1</appliedBootSheetIndex></heldItems>
