My textbook (《新实用汉语课本5》 p.16) gives the 累 in 拖累 a fourth tone: tuōlèi
but other dictionaries (e.g., CC-CEDICT) ascribe 累 a third tone: tuōlěi.
I feel like it's a bug in the textbook, but maybe there's more to this story, as it's not clear to me how third-tone 累 would be relevant to this word.
Question: Is the 累 in 拖累 third or fourth tone?
The historically more correct pronunciation is lei4. However, in 1985 mainland China produced a document called 普通话异读词审音表, which prescribed a single sound for some words that were pronounced differently by the people. This table had a tendency of 从俗, meaning taking whichever sound that was the more popular one, not the one that was historically correct.
The 1985 table says, 累 should be pronounced lei3, when it is used to mean 牵连 (to be involved, implicated), which it is in 连累.
In 2016 there was an attempt to revise the 1985 table. In the new table 累 in the sense of 牵连 was restored to lei4. However, this attempt seems to have been abandoned. The new table never became an official regulation.
The 1985 table: https://zh.wikisource.org/wiki/%E6%99%AE%E9%80%9A%E8%AF%9D%E5%BC%82%E8%AF%BB%E8%AF%8D%E5%AE%A1%E9%9F%B3%E8%A1%A8
辞源第三版 on the character 累. Note that the senses of 牵连, 连累 are ascribed to the pronunciation lei4.
tuōlěi is the modern Mainland pronunciation, tuōlèi the Taiwan one.
Etymologically, 累 as "to accumulate" was lěi (< ljweX), while "to involve", as here, was lèi (< ljweH). Currently, Mainland is shifting the second meaning to lěi as well, under the influence of the first.
In our real life 累 in 拖累 is in the fourth tone. The fourth tone is used when dealing with the meaning of tired. And the third tone is for those meaning like accumulation (积累).
累 pronounces with the 4th tone only when it means "tired".
To your question, I think 拖累 is /tuo1lei4/，拖累 means the same as 累 /lei4/. For example,
这孩子真累人. /lei4/
这孩子真拖累人. /lei4/
我被这孩子拖累着, 哪里也都去不了. /lei4/
There are different denotes in various dictionaries. For example:
《國語辭典》: 拖累(注音: ㄊㄨㄛㄌㄟˋ, 拼音: tuō lèi)
《现代汉语词典》: 拖累(拼音：tuō lěi, 注音：ㄊㄨㄛㄌㄟˇ）
I personally disagree with 《现代汉语词典》. The reason is that, in fact, 累 has 3 tones and each tone has a quite different basic meaning.
(1) 累 /lei2/ as in 累赘 /lei2zhui4/ or /lei2zhui0/, an extra burden. The meaning of /lei2/ is logically linked to /lei4/. I am not sure if it is a "change of tone due to tone 4 followed by another tone 4".
(2) 累 /lei3/ Accumulation, To add on it.
五年来这个基金累计有50%的收益.
他进城打工三年, 积累了一笔钱.
重复词：累累 /lei3lei3/ 这个恐怖组织罪恶累累. 秋天果树上硕果累累. (uncountable)
成语：危如累卵. /lei3/ one egg on top of another. 淡泊名利, 不为物累 /lei3/. Do not pursue the material interest. Do not set it on top of your objectives.
(3) 累 /lei4/ (1) (adj) tired. (我今天很累.) (2) (verb) to make someone feel tired, or someone is made tired. （你别累着了.）（这个工作不累.）(3) (verb) make trouble for someone. 你不要连累我们大家.
Sometimes people use them interchangeably. It may be due to local dialects, sound changes in a compound word, or other reasons.
