This is one of those questions that will ultimately depend on WHICH modern IPA you're looking at.

Ultimately, there are dozens of recipes, but the 19th century's IPA was still a high-hop pale ale. The currently trend of ultra-hoppy IPAs will in no way match up with the traditional recipe.

The Cascade hop is most commonly used in modern IPAs. It was first released in 1972. Tomahawk (another popular hops choice) is descended from Nugget, which was released in 1982. Even the Northern Brewer hops were bred in 1932.

IPA is pretty much a straight hops flavor (especially nowadays), so there would be some massive dissimilarities in flavor. Basically, different hops flavor and much more of it in the modern IPAs.

Looking at the malt ingredients, well, for a starter, liquid malt syrup seems to be a product of trying to get around Prohibition (really, there's some fascinating stuff there). Malt syrup does affect the quality of the final beverage, so any modern IPAs using syrup would taste slightly different (please note: I do not actually know if any commercial breweries use malt syrup).

Grain-wise, sure, there's some differences in the recipe, but let's be honest here, if you can taste the grain in a modern IPA, then you're not drinking an IPA. Yes, I'm biased. No, I don't much care for the super-IPAs that are flooding the craft beer market these days.

So, basically, 1800s IPA would be a hoppy pale ale, whereas a modern IPA can run the gamut from relatively like it to OH MY GOD I CAN ACTUALLY SEE THE HOPS.

Sources:

http://hopunion.com/cascade/ (And so on for hops history) http://www.kcet.org/living/food/the-public-kitchen/from-beer-to-cookies-malt-syrup-through-the-years.html (Date for malt syrup) http://beersmith.com/blog/2009/09/13/brewing-india-pale-ale-recipes-ipa-beer-styles/ (Number of homebrew recipes for IPAs)

I'm assuming most redditors are more familiar with American IPAs than English IPAs. So, would it taste like a fresh modern American IPA, bursting with citrus, pine, and tropical fruit flavors and aroma? Not at all.

First off, there's a big difference in my mind between modern American IPAs and modern English IPAs. To understand historical IPA, you must first familiarize yourself with the modern English IPA. From the BJCP style guidelines(.pdf) for American IPA: "Hop flavor...should reflect an American or New World hop character, such as citrus, floral, pine, resinous, spicy, tropical fruit, stone fruit, berry, melon, etc...Malt flavor...is generally clean..."

The guidelines for English IPA state: "The hop flavor should be similar to the aroma (floral, spicy-peppery, citrus-orange, and/or slightly grassy). Malt flavor should be...somewhat bready, optionally with light to medium-light biscuit-like, toasty, toffee-like and/or caramelly aspects."

Compared against each other: "[American] IPA, has less of the “English” character from malt, hops, and yeast (less caramel, bread, and toast; more American/New World hops than English; less yeast-derived esters), less body, and often has a more hoppy balance and is slightly stronger than most examples."

Historical IPAs brewed in England and exported to India were brewed with English hops such as East Kent Goldings, which are much more floral, herbal, and earthy compared to modern American IPA hops. They were hopped at much lower rates than modern IPAs. After brewing they were packaged into oak barrels and shipped across the tropics, possibly undergoing a secondary fermentation from wild yeast hiding out in the barrels. The beer would likely be served at cellar temperatures or warmer with low to moderate carbonation.

Much of the hoppiness would have faded due to age, oxidation, and high temperatures after such a voyage.

The tasting notes from a homebrew recipe meant to emulate the historical style describes the beer: "Even though I had the brew loaded with lots of hops in the kettle, after “warm-aging” it tasted more like a malty, heavy doppelbock than a mouth-tingling IPA. Yet, the brew also had a slightly acrid hop note at the very tail-end of the finish. Interestingly, the brew left totally different taste perceptions depending on the serving temperature. Only when drank chilled right out of the refrigerator, did the beer have a faint sharpness up-front. When served at room temperature, the predominant taste sensation was flat, almost insipid. However, when poured at cellar temperature, it tasted truly pleasant".

Here's another recipe, inspired by Mitch Steele's book IPA. Mitch Steele is the brewmaster at Stone.

Based on these descriptions and the brewing, fermentation, and aging process, just imagine an old, poorly stored oak-aged English IPA. Bitter but not hoppy. Maybe slightly funky. Almost like a session-strength English barleywine if such a style existed.

Edit: A great blog for reading about historical English styles is Shut up about Barclay Perkins. You can find a lot of recipes transcribed from the original logs of old breweries. But I have to emphasize how much of an impact the post-fermentation handling of a beer will have on its flavor. The recipes for a modern English IPA and a historical IPA might look similar, but recipes are only a small portion of what determines a beer's flavor. Brewing a top-notch IPA requires taking every precaution possible to limit oxygen exposure after fermentation. Then after dry-hopping the beer should be stored cold in a sanitized stainless steel container and consumed quickly. Beer sloshing about in an oak barrel on a ship crossing the equator? That beer will be unrecognizable from the original.