Of the major IP categories of patents, copyrights, and trademarks, trademarks have the most Biblical warrant. A trademark is a modern-day form of a seal, denoting a source of origin, and false marking of a trademark is really a kind of forgery. WLC 145 lists forgery as a violation of the 9th commandment; the proof text is 1 Kings 21:8, in which Jezebel misappropriates Ahab's hand and seal in her scheme to take Naboth's vineyard. It can be argued that false patent marking is a like kind of forgery.
So in response to OP's example of a counterfeit Nike shirt, the purpose of trademark is to identify source. A shirt with Nike's logo indicates to the buyer that it meets Nike's standards of quality, with its reputation or "goodwill" built up over 50 years. A manufacturer putting the logo on a lower-quality shirt is feeding off Nike's reputation, and when the shirt falls apart or fades after only two times in the laundry, it damages that reputation.
Copyright is a product of easy reproduction (e.g., printing press, photography) and patent a product of the Industrial Revolution -- you didn't need protection on your improved plow if it took the next blacksmith the same time and effort to make it as you did. The guild system also worked to protect manufacturing processes as trade secrets.
I believe patents are a permissible application of the general government mandate to praise those that do well (Romans 13:3, 1 Peter 2:14), particularly those who fulfill the creation mandate to subdue and take dominion over the earth (Genesis 1:28). This is not to say that there are abuses of the system, or that contemporary patent law is not in need of reform.
Copyright probably stands on the weakest ground -- the best justification I can think of immediately is an application of 1 Timothy 5:18. American copyright law is a mess as a collection of statutes written in response to each new form of reproducing and distributing content (all the way back to piano rolls) -- it really needs to be demolished and rebuilt from scratch.
In general, IP protection in its current form is more covenantal than moral (what is God's standard for a patent term?), a fifth commandment violation more than an eighth commandment violation. We may already be past the point where copyright can reasonably be enforced, and further innovations in 3D printing may be the tipping point against many patent categories.
Specifically with respect to industrial design, copyright and design patents are narrow. You can prevent somebody from making an exact replica of your table, as long as you meet creativity and novelty standards, but you can't prevent yourself from being the fountainhead of a whole design trend.