Never chase a lie. Let it alone, and it will run itself to death.

Lyman Beecher (1775-1863) American minister, preacher, abolitionist

(Attributed)

Where cited, it is usually to "Naval Journal (March 1848)," and in fact it does appear in The Sailor's Magazine and Naval Journal, Vol. 20, No. 7 (1848-03), but only as column filler, clearly not its original source. Such filler was sold to magazines and newspapers to fill out unused column space.

Nor is that the earliest appearance. Identically or very similarly formatted and attributed copies can be found in The New Jersey Freeman, Vol. 3, No. 5 (1848-02-05); the Ohio Cultivator, Vol. 3, No. 24 (1847-12-15); The Youth's Companion, Vol. 21, No. 30 (1847-11-25); the Boston Advent Herald, Vol. 14 (1847-11-13); the Christian Watchman, Vol. 28, No. 40 (1847-10-01).

I cannot find an original source for Beecher saying or writing this.

The quote (including its following "I can work out a good character much faster than any one can lie me out of it") can be found in other, slightly later, sources without attribution to Beecher (e.g., 1853, 1856). The Book of Thought (1858) attributes it to [Bishop George?] Berkeley.

A similar aphorism appears around the same time, unattributed, in Bronson, Elocution; or, Mental and Vocal Philosophy (1845) ("Never chase a lie; for if you keep quiet, truth will eventually overtake it") and in the American Agriculturalist, Vol. 11, No. 18 (1853-12-07) ("Never chase a lie; for if you be quiet, truth will eventually overtake it and destroy it").